nips nips2009 nips2009-31 knowledge-graph by maker-knowledge-mining

31 nips-2009-An LP View of the M-best MAP problem

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Author: Menachem Fromer, Amir Globerson

Abstract: We consider the problem of ﬁnding the M assignments with maximum probability in a probabilistic graphical model. We show how this problem can be formulated as a linear program (LP) on a particular polytope. We prove that, for tree graphs (and junction trees in general), this polytope has a particularly simple form and diﬀers from the marginal polytope in a single inequality constraint. We use this characterization to provide an approximation scheme for non-tree graphs, by using the set of spanning trees over such graphs. The method we present puts the M -best inference problem in the context of LP relaxations, which have recently received considerable attention and have proven useful in solving diﬃcult inference problems. We show empirically that our method often ﬁnds the provably exact M best conﬁgurations for problems of high tree-width. A common task in probabilistic modeling is ﬁnding the assignment with maximum probability given a model. This is often referred to as the MAP (maximum a-posteriori) problem. Of particular interest is the case of MAP in graphical models, i.e., models where the probability factors into a product over small subsets of variables. For general models, this is an NP-hard problem [11], and thus approximation algorithms are required. Of those, the class of LP based relaxations has recently received considerable attention [3, 5, 18]. In fact, it has been shown that some problems (e.g., ﬁxed backbone protein design) can be solved exactly via sequences of increasingly tighter LP relaxations [13]. In many applications, one is interested not only in the MAP assignment but also in the M maximum probability assignments [19]. For example, in a protein design problem, we might be interested in the M amino acid sequences that are most stable on a given backbone structure [2]. In cases where the MAP problem is tractable, one can devise tractable algorithms for the M best problem [8, 19]. Speciﬁcally, for low tree-width graphs, this can be done via a variant of max-product [19]. However, when ﬁnding MAPs is not tractable, it is much less clear how to approximate the M best case. One possible approach is to use loopy max-product to obtain approximate max-marginals and use those to approximate the M best solutions [19]. However, this is largely a heuristic and does not provide any guarantees in terms of optimality certiﬁcates or bounds on the optimal values. LP approximations to MAP do enjoy such guarantees. Speciﬁcally, they provide upper bounds on the MAP value and optimality certiﬁcates. Furthermore, they often work for graphs with large tree-width [13]. The goal of the current work is to leverage the power of LP relaxations to the M best case. We begin by focusing on the problem of ﬁnding the second best solution. We show how it can be formulated as an LP over a polytope we call the “assignment-excluding marginal polytope”. In the general case, this polytope may require an exponential number of inequalities, but we prove that when the graph is a tree it has a very compact representation. We proceed to use this result to obtain approximations to the second best problem, and show how these can be tightened in various ways. Next, we show how M best assignments can be found by relying on algorithms for 1 second best assignments, and thus our results for the second best case can be used to devise an approximation algorithm for the M best problem. We conclude by applying our method to several models, showing that it often ﬁnds the exact M best assignments. 1 The M-best MAP problem and its LP formulation Consider a function on n variables deﬁned as: f (x1 , . . . , xn ; θ) = θij (xi , xj ) + ij∈E θi (xi ) (1) i∈V where V and E are the vertices and nodes of a graph G with n nodes. We shall be interested in the M assignments with largest f (x; θ) value.1 Denote these by x(1) , . . . , x(M) , so that x(1) is the assignment that maximizes f (x; θ), x(2) is the 2nd best assignment, etc. The MAP problem (i.e., ﬁnding x(1) ) can be formulated as an LP as follows [15]. Let µ be a vector of distributions that includes {µij (xi , xj )}ij∈E over edge variables and {µi (xi )}i∈V over nodes. The set of µ that arise from some joint distribution is known as the marginal polytope [15] and is denoted by M(G). Formally: M(G) = {µ | ∃p(x) ∈ ∆ s.t. p(xi , xj ) = µij (xi , xj ) , p(xi ) = µi (xi )} . where ∆ is the set of distributions on x. The MAP problem can then be shown to be equivalent to the following LP:2 max f (x; θ) = max µ · θ , (2) x µ∈M(G) It can be shown that this LP always has a maximizing µ that is a vertex of M(G) and is integral. Furthermore, this µ corresponds to the MAP assignment x(1) . Although the number of variables in this LP is only O(|E| + |V |), the diﬃculty comes from an exponential number of linear inequalities generally required to describe the marginal polytope M(G). We shall ﬁnd it useful to deﬁne a mapping between assignments x and integral vertices of the polytope. Given an integral vertex v ∈ M(G), deﬁne x(v) to be the assignment that maximizes vi (xi ). And, given an assignment z deﬁne v(z) to be the integral vertex in M(G) corresponding to the assignment z. Thus the LP in Eq. 2 will be maximized by v(x(1) ). One simple outer bound of the marginal polytope is the local polytope ML (G), which only enforces pairwise constraints between variables:     µi (xi ) = 1 (3) µij (xi , xj ) = µj (xj ), µij (xi , xj ) = µi (xi ), ML (G) = µ ≥ 0   x x x i j i The LP relaxation is then to maximize µ · θ where µ ∈ ML (G). For tree structured graphs, ML (G) = M(G) [15] and thus the LP relaxation yields the exact MAP x(1) . An LP Formulation for the 2nd -best MAP 2 Assume we found the MAP assignment x(1) and are now interested in ﬁnding x(2) . Is there a simple LP whose solution yields x(2) ? We begin by focusing on the case where G is a tree so that the local LP relaxation is exact. We ﬁrst treat the case of a connected tree. To construct an LP whose solution is x(2) , a natural approach is to use the LP for x(1) (i.e., the LP in Eq. 2) but somehow eliminate the solution x(1) using additional constraints. This, however, is somewhat trickier than it sounds. The key diﬃculty is that the new constraints should not generate fractional vertices, so that the resulting LP is still exact. We begin by deﬁning the polytope over which we need to optimize in order to obtain x(2) . 1 2 This is equivalent to ﬁnding P maximum probability assignments for a model p(x) ∝ ef (x;θ) . the P P P We use the notation µ · θ = ij∈E xi ,xj µij (xi , xj )θij (xi , xj ) + i xi µi (xi )θi (xi ) 2 Deﬁnition 1. The assignment-excluding marginal polytope is deﬁned as: ˆ M(G, z) = {µ | ∃p(x) ∈ ∆ s.t. p(z) = 0, p(xi , xj ) = µij (xi , xj ), p(xi ) = µi (xi )} . ˆ M(G, z) is simply the convex hull of all (integral) vectors v(x) for x = z. (4) ˆ The following result shows that optimizing over M(G, x(1) ) will yield the second best soluˆ tion x(2) , so that we refer to M(G, x(1) ) as the second-best marginal polytope. Lemma 1. The 2nd best solution is obtained via the following LP: maxx=x(1) f (x; θ) = maxµ∈M(G,x(1) ) µ · θ. Furthermore, the µ that maximizes the LP on ˆ the right is integral and corresponds to the second-best MAP assignment x(2) . The proof is similar to that of Eq. 2: instead of optimizing over x, we optimize over distributions p(x), while enforcing that p(x(1) ) = 0 so that x(1) is excluded from the maximization. The key question which we now address is how to obtain a simple characterization of ˆ ˆ M(G, z). Intuitively, it would seems that M(G, z) should be “similar” to M(G), such that it can be described as M(G) plus some constraints that “block” the assignment z. To illustrate the diﬃculty in ﬁnding such “blocking” constraints, consider the following constraint, originally suggested by Santos [10]: i µi (zi ) ≤ n − 1. This inequality is not satisﬁed by µ = v(z) since v(z) attains the value n for the LHS of the above. Furthermore, for any x = z and µ = v(x), the LHS would be n − 1 or less. Thus, this inequality separates ˆ v(z) from all other integral vertices. One might conclude that we can deﬁne M(G, z) by adding this inequality to M(G). The diﬃculty is that the resulting polytope has fractional vertices,3 and maximizing over it won’t generally yield an integral solution. It turns out that there is a diﬀerent inequality that does yield an exact characterization of ˆ M(G, z) when G is a tree. We now deﬁne this inequality and state our main theorem. Deﬁnition 2. Consider the functional I(µ, z) (which is linear in µ): (1 − di )µi (zi ) + I(µ, z) = i µij (zi , zj ) (5) ij∈E where di is the degree of node i in the tree graph G. ˆ Theorem 1. Adding the single inequality I(µ, z) ≤ 0 to M(G) yields M(G, z). ˆ M(G, z) = {µ | µ ∈ M(G), I(µ, z) ≤ 0 } (6) The theorem is proved in the appendix. Taken together with Lemma 1, it implies that x(2) may be obtained via an LP that is very similar to the MAP-LP, but has an additional constraint. We note the interesting similarity between I(µ, z) and the Bethe entropy [20]. The only diﬀerence is that in Bethe, µi , µij are replaced by H(Xi ), H(Xi , Xj ) respectively.4 The theorem also generalizes to the case where G is not a tree, but we have a junction tree for G. In this case, the theorem still holds if we deﬁne a generalized I(µ, z) inequality as: (1 − dS )µS (zS ) + S∈S µC (zC ) ≤ 0 (7) C∈C where C and S are the junction tree cliques and their separators, respectively, and dS is the number of cliques that intersect on separator S. In this case, the marginal polytope should enforce consistency between marginals µC (zC ) and their separators µS (zS ). However, such a characterization requires variables whose cardinality is exponential in the tree-width and is thus tractable only for graphs of low tree-width. In the next section, we address approximations for general graphs. A corresponding result exists for the case when G is a forest. In this case, the inequality in Eq. 6 is modiﬁed to: I(µ, z) ≤ |P | − 1, where |P | denotes the number of connected components of G. Interestingly, for a graph without edges, this gives the Santos inequality. 3 Consider the case of a single edge between 2 nodes where the MAP assignment is (0, 0). Adding the inequality µ1 (0) + µ2 (0) ≤ 1 produces the fractional vertex (0.5, 0.5). 4 The connection to Bethe can be more clearly understood from a duality-based proof of Theorem 1. We will cover this in an extended version of the manuscript. 3 2nd best LPs for general graphs - Spanning tree inequalities 3 When the graph G is not a tree, the marginal polytope M(G) generally requires an exponential number of inequalities. However, as mentioned above, it does have an exact description in terms of marginals over cliques and separators of a junction tree. Given such marginals on ˆ junction tree cliques, we also have an exact characterization of M(G, z) via the constraint in Eq. 7. However, in general, we cannot aﬀord to be exponential in tree-width. Thus a common strategy [15] is to replace M(G) with an outer bound that enforces consistency between marginals on overlapping sets of variables. The simplest example is ML (G) in Eq. 3. ˆ In what follows, we describe an outer-bound approximation scheme for M(G, z). We use ML (G) as the approximation for M(G) (more generally ML (G) can enforce consistency between any set of small regions, e.g., triplets). When G is not a tree, the linear constraint in ˆ Eq. 6 will no longer suﬃce to derive M(G, z). Moreover, direct application of the inequality will incorrectly remove some integral vertices. An alternative approach is to add inequalities that separate v(z) from the other integral vertices. This will serve to eliminate more and more fractional vertices, and if enough constraints are added, this may result in an integral solution. One obvious family of such constraints are those corresponding to spanning trees in G and have the form of Eq. 5. Deﬁnition 3. Consider any T that is a spanning tree of G. Deﬁne the functional I T (µ, z): (1 − dT )µi (zi ) + i I T (µ, z) = i µij (zi , zj ) (8) ij∈T where dT is the degree of i in T . We refer to I T (µ, z) ≤ 0 as a spanning tree inequality. i For any sub-tree T of G, the corresponding spanning tree inequality separates the vertex v(z) from the other vertices. This can be shown via similar arguments as in the proof of Theorem 1. Note, however, that the resulting polytope may still have fractional vertices. The above argument shows that any spanning tree provides a separating inequality for ˆ M(G, z). In principle, we would like to use as many such inequalities as possible. Deﬁnition 4. The spanning tree assignment-excluding marginal polytope is deﬁned as: ˆ MST (G, z) = µ | µ ∈ ML (G), L ∀ tree T ⊆ E I T (µ, z) ≤ 0 (9) where the ST notation indicates the inclusion of all spanning tree inequalities for G.5 Thus, we would actually like to perform the following optimization problem: max ˆ µ∈MST (G,z) L µ·θ ˆ as an approximation to optimization over M(G, z); i.e., we seek the optimal µ subject to all spanning tree inequalities for G with the ambition that this µ be integral and thus provide the non-z MAP assignment, with a certiﬁcate of optimality. Although the number of spanning trees is exponential in n, it turns out that all spanning inequalities can be used in practice. One way to achieve this is via a cutting plane algorithm [12] that ﬁnds the most violated spanning tree inequality and adds it to the LP. To implement this eﬃciently, we note that for a particular µ and a spanning tree T , the value of I T (µ, z) can be decomposed into a sum over the edges in T (and a T -independent constant): I T (µ, z) = µi (zi ) µij (zi , zj ) − µi (zi ) − µj (zj ) + (10) i ij∈T The tree maximizing the above is the maximum-weight spanning tree with edge-weights wij = µij (zi , zj ) − µi (zi ) − µj (zj ). It can thus be found eﬃciently. The cutting plane algorithm proceeds as follows. We start by adding an arbitrary spanning tree. Then, as long as the optimal µ is fractional, we ﬁnd the spanning tree inequality that µ most violates (where this is implemented via the maximum-weight spanning tree). This constraint will necessarily remove µ from the polytope. If there are no violated inequalities 5 ˆ ˆL Note that M(G, z) ⊆ MST (G, z) ⊂ ML (G). 4 but µ is still fractional, then spanning tree inequalities do not suﬃce to ﬁnd an integral solution (but see below on hypertree constraints to add in this case). In practice, we found that only a relatively small number of inequalities are needed to successfully yield an integral solution, or determine that all such inequalities are already satisﬁed. An alternative approach for solving the all spanning-tree problem is to work via the dual. The dual variables roughly correspond to points in the spanning tree polytope [16], optimization over which can be done in polynomial time, e.g., via the ellipsoid algorithm. We do not pursue this here since the cutting plane algorithm performed well in our experiments. ˆ As mentioned earlier, we can exactly characterize M(G, z) using Eq. 7, albeit at a cost exponential in the tree-width of the graph. A practical compromise would be to use inequalities over clique trees of G, where the cliques are relatively small, e.g., triplets. The corresponding constraint (Eq. 7 with the small cliques and their separators) will necessarily separate v(z) from the other integral vertices. Finding the maximally violated such inequality is an NP-hard problem, equivalent to a prize collecting Steiner tree problem, but recent work has found that such problems are often exactly solvable in practice [7]. It thus might be practical to include all such trees as constraints using a cutting plane algorithm. 4 From 2nd -best to M-best Thus far, we only dealt with the 2nd best case. As we show now, it turns out that the 2nd -best formalism can be used to devise an algorithm for M best. We begin by describing an algorithm for the exact M best and then show how it can be used to approximate those via the approximations for 2nd best described above. Fig. 1 describes our scheme, which we call Partitioning for Enumerating Solutions (or PES) for solving the M best problem. The scheme is general and only assumes that MAP-“like” problems can be solved. It is inspired by several pre-existing M best solution schemes [4, 6, 8, 19] but diﬀers from them in highlighting the role of ﬁnding a second best solution within a given subspace. for m ← 1 to M do if m = 1 then Run MAP solver to obtain the best assignment: x(1) ≡ arg max f (x; θ) CONSTRAINTS1 ← ∅ else k ←− arg max ′ k′ ∈{1,...,m−1} f (y(k ) ; θ) // sub-space containing mth best assignment x(m) ← y(k) // mth best assignment // A variable choice that distinguishes x(m) from x(k) : (m) (v, a) ← any member of the set {(i, xi (m) ) : xi (k) = xi } CONSTRAINTSm ← CONSTRAINTSk ∪ {xv = a} // Eliminate x(k) (as MAP) from subspace m CONSTRAINTSk ← CONSTRAINTSk ∪ {xv = a} // Eliminate x(m) (as 2nd -best) from subspace k y(k) ← CalcNextBestSolution(CONSTRAINTSk , x(k) ) end y(m) ← CalcNextBestSolution(CONSTRAINTSm , x(m) ) end return {x(m) }M m=1 /* Find next best solution in sub-space defined by CONSTRAINTS */ Function CalcNextBestSolution(CONSTRAINTS, x(∗) ) // x(∗) is the MAP in the sub-space defined by CONSTRAINTS: Run MAP solver to obtain the second-best solution: y ≡ arg max f (x; θ), and return y. x=x(∗) ,CONSTRAINTS end Figure 1: Pseudocode for the PES algorithm. The modus operandi of the PES algorithm is to eﬃciently partition the search space while systematically excluding all previously determined assignments. Signiﬁcantly, any MAP 5 Attractive Grids Ranks Run-times 1 50 Mixed Grids Ranks Run-times 1 50 0.5 0 S N B 0 Hard Protein SCP Ranks Run-times 1 50 0.5 S N B 0 0 S+R N+R B+R 0.5 S+R N+R B+R 0 S+R B B+R 0 S+R B B+R Figure 2: Number of best ranks and normalized run-times for the attractive and mixed grids, and the more diﬃcult protein SCP problems. S, N, and B denote the STRIPES, Nilsson, and BMMF algorithms. Algorithms marked with +R denote that regions of variables were added for those runs. solver can be plugged into it, on the condition that it is capable of solving the arg max in the CalcNextBestSolution subroutine. The correctness of PES can be shown by observing that at the M th stage, all previous best solutions are excluded from the optimization and no other assignment is excluded. Of note, this simple partitioning scheme is possible due to the observation that the ﬁrst-best and second-best MAP assignments must diﬀer in the assignment of at least one variable in the graph. The main computational step of the PES algorithm is to maximize f (x; θ) subject to x = x(∗) and x ∈ CONSTRAINTS (see the CalcNextBestSolution subroutine). The CONSTRAINTS set merely enforces that some of the coordinates of x are either equal to or diﬀerent from speciﬁed values.6 Within the LP, these can be enforced by setting µi (xi = a) = 1 or µi (xi = a) = 0. It can be shown that if one optimizes µ · θ with ˆ these constraints and µ ∈ M(G, x(∗) ), the solution is integral. Thus, the only element ˆ requiring approximation in the general case is the description of M(G, x(∗) ). We choose as ˆ this approximation the polytope MST (G, x(∗) ) in Eq. 9. We call the resulting approximaL tion algorithm Spanning TRee Inequalities and Partitioning for Enumerating Solutions, or STRIPES. In the next section, we evaluate this scheme experimentally. 5 Experiments We compared the performance of STRIPES to the BMMF algorithm [19] and the Lawler/Nilsson algorithm [6, 8]. Nilsson’s algorithm is equivalent to PES where the 2nd best assignment is obtained from maximizations within O(n) partitions, so that its runtime is O(n) times the cost of ﬁnding a single MAP. Here we approximated each MAP with its LP relaxation (as in STRIPES), so that both STRIPES and Nilsson come with certiﬁcates of optimality when their LP solutions are integral. BMMF relies on loopy BP to approximate the M best solutions.7 We used M = 50 in all experiments. To compare the algorithms, we pooled all their solutions, noting the 50 top probabilities, and then counted the fraction of these that any particular algorithm found (its solution rank). For run-time comparisons, we normalized the times by the longest-running algorithm for each example. We begin by considering pairwise MRFs on binary grid graphs of size 10 × 10. In the ﬁrst experiment, we used an Ising model with attractive (submodular) potentials, a setting in which the pairwise LP relaxation is exact [14]. For each grid edge ij, we randomly chose Jij ∈ [0, 0.5], and local potentials were randomized in the range ±0.5. The results for 25 graphs are shown in Fig. 2. Both the STRIPES and Nilsson algorithms obtained the 50 optimal solutions (as learned from their optimality certiﬁcates), while BMMF clearly fared less well for some of the graphs. While the STRIPES algorithm took < 0.5 to 2 minutes to run, the Nilsson algorithm took around 13 minutes. On the other hand, BMMF was quicker, taking around 10 seconds per run, while failing to ﬁnd a signiﬁcant portion of the top solutions. Overall, the STRIPES algorithm was required to employ up to 19 spanning tree inequalities per calculation of second-best solution. 6 This is very diﬀerent from the second best constraint, since setting x1 = 1 blocks all assignments with this value, as opposed to setting x = 1 which blocks only the assignment with all ones. 7 For BMMF, we used the C implementation at http://www.cs.huji.ac.il/~ talyam/ inference.html. The LPs for STRIPES and Nilsson were solved using CPLEX. 6 Next, we studied Ising models with mixed interaction potentials (with Jij and the local potentials randomly chosen in [−0.5, 0.5]). For almost all of the 25 models, all three algorithms were not able to successfully ﬁnd the top solutions. Thus, we added regions of triplets (two for every grid face) to tighten the LP relaxation (for STRIPES and Nilsson) and to perform GBP instead of BP (for BMMF). This resulted in STRIPES and Nilsson always provably ﬁnding the optimal solutions, and BMMF mostly ﬁnding these solutions (Fig. 2). For these more diﬃcult grids, however, STRIPES was the fastest of the algorithms, taking 0.5 - 5 minutes. On the other hand, the Nilsson and BMMF algorithms took 18 minutes and 2.5 7 minutes, respectively. STRIPES added up to 23 spanning tree inequalities per iteration. The protein side-chain prediction (SCP) problem is to to predict the placement of amino acid side-chains given a protein backbone [2, 18]. Minimization of a protein energy function corresponds to ﬁnding a MAP assignment for a pairwise MRF [19]. We employed the dataset of [18] (up to 45 states per variable, mean approximate tree-width 50), running all algorithms to calculate the optimal side-chain conﬁgurations. For 315 of 370 problems in the dataset, the ﬁrst MAP solution was obtained directly as a result of the LP relaxation having an integral solution (“easy” problems). STRIPES provably found the subsequent top 50 solutions within 4.5 hours for all but one of these cases (up to 8 spanning trees per calculation), and BMMF found the same 50 solutions for each case within 0.5 hours; note that only STRIPES provides a certiﬁcate of optimality for these solutions. On the other hand, only for 146 of the 315 problems was the Nilsson method able to complete within ﬁve days; thus, we do not compare its performance here. For the remaining 55 (“hard”) problems (Fig. 2), we added problem-speciﬁc triplet regions using the MPLP algorithm [13]. We then ran the STRIPES algorithm to ﬁnd the optimal solutions. Surprisingly, it was able to exactly ﬁnd the 50 top solutions for all cases, using up to 4 standard spanning tree inequalities per second-best calculation. The STRIPES run-times for these problems ranged from 6 minutes to 23 hours. On the other hand, whether running BMMF without these regions (BP) or with the regions (GBP), it did not perform as well as STRIPES in terms of the number of high-ranking solutions or its speed. To summarize, STRIPES provably found the top 50 solutions for 369 of the 370 protein SCP problems. 6 Conclusion ˆ In this work, we present a novel combinatorial object M(G, z) and show its utility in obtaining the M best MAP assignments. We provide a simple characterization of it for tree structured graphs, and show how it can be used for approximations in non-tree graphs. As with the marginal polytope, many interesting questions arise about the properties of ˆ M(G, z). For example, in which non-tree cases can we provide a compact characterization (e.g., as for the cut-polytope for planar graphs [1]). Another compelling question is in which problems the spanning tree inequalities are provably optimal. An interesting generalization of our method is to predict diverse solutions satisfying some local measure of “distance” from each other, e.g., as in [2]. Here we studied the polytope that results from excluding one assignment. An intriguing question is to characterize the polytope that excludes M assignments. We have found that it does not simply correspond to adding M constraints I(µ, z i ) ≤ 0 for i = 1, . . . , M , so its ˆ geometry is apparently more complicated than that of M(G, z). Here we used LP solvers to solve for µ. Such generic solvers could be slow for large-scale problems. However, in recent years, specialized algorithms have been suggested for solving MAP-LP relaxations [3, 5, 9, 17]. These use the special form of the constraints to obtain local-updates and more scalable algorithms. We intend to apply these schemes to our method. Finally, our empirical results show that our method indeed leverages the power of LP relaxations and yields exact M best optimal solutions for problems with large tree-width. Acknowledgements We thank Nati Linial for his helpful discussions and Chen Yanover and Talya Meltzer for their insight and help in running BMMF. We also thank the anonymous reviewers for their useful advice. 7 A Proof of Theorem 1 Recall that for any µ ∈ M(G), there exists a probability density p(x) s.t. µ = x p(x)v(x). Denote pµ (z) as the minimal value of p(z) among all p(x) that give µ. We prove that ˆ pµ (z) = max(0, I(µ, z)), from which the theorem follows (since pµ (z) = 0 iﬀ µ ∈ M(G, z)). The proof is by induction on n. For n = 1, the node has degree 0, so I(µ, z) = µ1 (z1 ). Clearly, pµ (z) = µ1 (z1 ), so pµ (z) = I(µ, z). For n > 1, there must exist a leaf in G ˆ (assume that its index is n and its neighbor’s is n − 1). Denote G as the tree obtained ˆ by removing node n and its edge with n − 1. For any assignment x, denote x as the corresponding sub-assignment for the ﬁrst n − 1 variables. Also, any µ can be derived by ˆ ˆ adding appropriate coordinates to a unique µ ∈ M(G). For an integral vertex µ = v(x), ˆˆ ˆ ˆ ˆ ˆ x denote its projected µ as v (ˆ ). Denote by I(µ, z ) the functional in Eq. 5 applied to G. For ˆ any µ and its projected µ, it can be seen that: ˆˆ ˆ I(µ, z) = I(µ, z ) − α (11) where we deﬁne α = xn =zn µn−1,n (zn−1 , xn ) (so 0 ≤ α ≤ 1). The inductive assumption ˆ ˆ ˆ gives a p(ˆ ) that has marginals µ and also p(ˆ ) = max(0, I(µ, z )). We next use p(ˆ ) to ˆx ˆz ˆx construct a p(x) that has marginals µ and the desired minimal pµ (z). Consider three cases: ˆˆ ˆ I. I(µ, z) ≤ 0 and I(µ, z ) ≤ 0. From the inductive assumption, pµ (ˆ ) = 0, so we deﬁne: ˆˆ z µn−1,n (xn−1 , xn ) p(x) = p(ˆ ) ˆx (12) µn−1 (xn−1 ) which indeed marginalizes to µ, and p(z) = 0 so that pµ (z) = 0 as required. If µn−1 (xn−1 ) = 0, then p(ˆ ) is necessarily 0, in which case we deﬁne p(x) = 0. Note that this construction ˆx is identical to that used in proving that ML (G) = M(G) for a tree graph G. ˆˆ ˆ II. I(µ, z) > 0. Based on Eq. 11 and α ≥ 0, we have I(µ, z ) > 0. Applying the inductive ˆ µ, z ) = pµ (ˆ ) > 0. Now, deﬁne p(x) so that p(z) = I(µ, z): ˆ assumption to µ, we obtain I( ˆ ˆ ˆˆ z xl , l ≤ n − 2 δ(xn−1 = zn−1 ) δ(xn = zn ) p(x) no constraint 0 no constraint As in Eq. 12 0 0 ∃ l x l = zl 1 ∀ l x l = zl 1 µn−1,n (zn−1 , xn ) 1 1 p(ˆ ) ˆx 0 I(µ, z) Simple algebra shows that p(x) is non-negative and has µ as marginals. We now show that p(z) is minimal. Based on the inductive assumption and Eq. 11, it can easily be shown that I(v(z), z) = 1, I(v(x), z) ≤ 0 for x = z. For any p(x) s.t. µ = x p(x)v(x), from linearity, I(µ, z) = p(z) + x=z p(x)I(v(x), z) ≤ p(z) (since I(v(x), z) ≤ 0 for x = z). Since the p(z) we deﬁne achieves this lower bound, it is clearly minimal. ˆˆ ˆ ˆ III. I(µ, z) ≤ 0 but I(µ, z ) > 0. Applying the inductive assumption to µ, we see that ˆ µ, z ) > 0; Eq. 11 implies α − I(µ, z ) ≥ 0. Deﬁne β = µn−1 (zn−1 ) − pµ (ˆ ), which ˆˆ ˆ ˆˆ z pµ (ˆ ) = I( ˆ ˆ ˆˆ z ˆ is non-negative since µn−1 (zn−1 ) = µn−1 (ˆ n−1 ) and p marginalizes to µ. Deﬁne p(x) as: ˆ z ˆ xl , l ≤ n − 2 δ(xn−1 = zn−1 ) δ(xn = zn ) no constraint 0 no constraint ∃ l x l = zl As in Eq. 12 0 ˆ ˆ z µ (z ,x ) p(ˆ ) n−1,n βn−1 n α−I(µ,ˆ ) ˆx α µ (z ,z ) p(ˆ ) n−1,n βn−1 n ˆx (z ,x ) ˆˆ ˆ µ I(µ, z ) n−1,n αn−1 n 1 0 0 1 1 ∀ l x l = zl p(x) 1 which indeed marginalizes to µ, and p(z) = 0 so that pµ (z) = 0, as required. 8 References [1] F. Barahona. On cuts and matchings in planar graphs. Math. Program., 60(1):53–68, 1993. [2] M. Fromer and C. Yanover. Accurate prediction for atomic-level protein design and its application in diversifying the near-optimal sequence space. Proteins: Structure, Function, and Bioinformatics, 75:682–705, 2009. [3] A. Globerson and T. Jaakkola. Fixing max-product: Convergent message passing algorithms for MAP LP-relaxations. In J. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 21. MIT Press, Cambridge, MA, 2007. [4] E. Kloppmann, G. M. Ullmann, and T. Becker. An extended dead-end elimination algorithm to determine gap-free lists of low energy states. Journal of Comp. Chem., 28:2325–2335, 2007. [5] N. Komodakis and N. Paragios. Beyond loose LP-relaxations: Optimizing MRFs by repairing cycles. In D. Forsyth, P. Torr, and A. Zisserman, editors, ECCV, pages 806–820, Heidelberg, Germany, 2008. Springer. [6] E. L. Lawler. A procedure for computing the K best solutions to discrete optimization problems and its application to the shortest path problem. Management Science, 18(7):401–405, 1972. [7] I. Ljubic, R. Weiskircher, U. Pferschy, G. W. Klau, P. Mutzel, and M. Fischetti. An algorithmic framework for the exact solution of the prize-collecting steiner tree problem. Mathematical Programming, 105:427–449, Feb 2006. [8] D. Nilsson. An eﬃcient algorithm for ﬁnding the M most probable conﬁgurations in probabilistic expert systems. Statistics and Computing, 8:159–173, Jun 1998. [9] P. Ravikumar, A. Agarwal, and M. Wainwright. Message-passing for graph-structured linear programs: proximal projections, convergence and rounding schemes. In Proc. of the 25th international conference on Machine learning, pages 800–807, New York, NY, USA, 2008. ACM. [10] E. Santos. On the generation of alternative explanations with implications for belief revision. In Proc. of the 7th Annual Conference on Uncertainty in Artiﬁcial Intelligence, 1991. [11] Y. Shimony. Finding the MAPs for belief networks is NP-hard. 68(2):399–410, 1994. Aritiﬁcal Intelligence, [12] D. Sontag and T. Jaakkola. New outer bounds on the marginal polytope. In J. Platt, D. Koller, Y. Singer, and S. Roweis, editors, Advances in Neural Information Processing Systems 20, pages 1393–1400. MIT Press, Cambridge, MA, 2007. [13] D. Sontag, T. Meltzer, A. Globerson, T. Jaakkola, and Y. Weiss. Tightening LP relaxations for MAP using message passing. In Proc. of the 24th Annual Conference on Uncertainty in Artiﬁcial Intelligence, pages 503–510, 2008. [14] B. Taskar, S. Lacoste-Julien, and M. I. Jordan. Structured prediction, dual extragradient and bregman projections. J. Mach. Learn. Res., 7:1627–1653, 2006. [15] M. Wainwright and M. Jordan. Graphical models, exponential families, and variational inference. Found. Trends Mach. Learn., 1(1-2):1–305, 2008. [16] M. J. Wainwright, T. Jaakkola, and A. S. Willsky. A new class of upper bounds on the log partition function. IEEE Transactions on Information Theory, 51(7):2313–2335, 2005. [17] T. Werner. A linear programming approach to max-sum problem: A review. IEEE Trans. Pattern Anal. Mach. Intell., 29(7):1165–1179, 2007. [18] C. Yanover, T. Meltzer, and Y. Weiss. Linear programming relaxations and belief propagation – an empirical study. Journal of Machine Learning Research, 7:1887–1907, 2006. [19] C. Yanover and Y. Weiss. Finding the M most probable conﬁgurations using loopy belief propagation. In Advances in Neural Information Processing Systems 16. MIT Press, Cambridge, MA, 2004. [20] J. Yedidia, W. W.T. Freeman, and Y. Weiss. Constructing free-energy approximations and generalized belief propagation algorithms. IEEE Trans. on Information Theory, 51(7):2282– 2312, 2005. 9

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 We prove that, for tree graphs (and junction trees in general), this polytope has a particularly simple form and diﬀers from the marginal polytope in a single inequality constraint. [sent-6, score-1.077]

2 We use this characterization to provide an approximation scheme for non-tree graphs, by using the set of spanning trees over such graphs. [sent-7, score-0.413]

3 We show empirically that our method often ﬁnds the provably exact M best conﬁgurations for problems of high tree-width. [sent-9, score-0.142]

4 A common task in probabilistic modeling is ﬁnding the assignment with maximum probability given a model. [sent-10, score-0.136]

5 , ﬁxed backbone protein design) can be solved exactly via sequences of increasingly tighter LP relaxations [13]. [sent-19, score-0.253]

6 In many applications, one is interested not only in the MAP assignment but also in the M maximum probability assignments [19]. [sent-20, score-0.203]

7 For example, in a protein design problem, we might be interested in the M amino acid sequences that are most stable on a given backbone structure [2]. [sent-21, score-0.208]

8 One possible approach is to use loopy max-product to obtain approximate max-marginals and use those to approximate the M best solutions [19]. [sent-25, score-0.162]

9 The goal of the current work is to leverage the power of LP relaxations to the M best case. [sent-30, score-0.158]

10 We show how it can be formulated as an LP over a polytope we call the “assignment-excluding marginal polytope”. [sent-32, score-0.315]

11 In the general case, this polytope may require an exponential number of inequalities, but we prove that when the graph is a tree it has a very compact representation. [sent-33, score-0.514]

12 Next, we show how M best assignments can be found by relying on algorithms for 1 second best assignments, and thus our results for the second best case can be used to devise an approximation algorithm for the M best problem. [sent-35, score-0.314]

13 , xn ; θ) = θij (xi , xj ) + ij∈E θi (xi ) (1) i∈V where V and E are the vertices and nodes of a graph G with n nodes. [sent-40, score-0.174]

14 , x(M) , so that x(1) is the assignment that maximizes f (x; θ), x(2) is the 2nd best assignment, etc. [sent-45, score-0.189]

15 The set of µ that arise from some joint distribution is known as the marginal polytope [15] and is denoted by M(G). [sent-50, score-0.315]

16 p(xi , xj ) = µij (xi , xj ) , p(xi ) = µi (xi )} . [sent-53, score-0.13]

17 Furthermore, this µ corresponds to the MAP assignment x(1) . [sent-56, score-0.136]

18 Although the number of variables in this LP is only O(|E| + |V |), the diﬃculty comes from an exponential number of linear inequalities generally required to describe the marginal polytope M(G). [sent-57, score-0.513]

19 We shall ﬁnd it useful to deﬁne a mapping between assignments x and integral vertices of the polytope. [sent-58, score-0.226]

20 Given an integral vertex v ∈ M(G), deﬁne x(v) to be the assignment that maximizes vi (xi ). [sent-59, score-0.317]

21 And, given an assignment z deﬁne v(z) to be the integral vertex in M(G) corresponding to the assignment z. [sent-60, score-0.453]

22 For tree structured graphs, ML (G) = M(G) [15] and thus the LP relaxation yields the exact MAP x(1) . [sent-64, score-0.309]

23 An LP Formulation for the 2nd -best MAP 2 Assume we found the MAP assignment x(1) and are now interested in ﬁnding x(2) . [sent-65, score-0.136]

24 We begin by focusing on the case where G is a tree so that the local LP relaxation is exact. [sent-67, score-0.273]

25 The key diﬃculty is that the new constraints should not generate fractional vertices, so that the resulting LP is still exact. [sent-74, score-0.151]

26 We begin by deﬁning the polytope over which we need to optimize in order to obtain x(2) . [sent-75, score-0.258]

27 the P P P We use the notation µ · θ = ij∈E xi ,xj µij (xi , xj )θij (xi , xj ) + i xi µi (xi )θi (xi ) 2 Deﬁnition 1. [sent-77, score-0.298]

28 The assignment-excluding marginal polytope is deﬁned as: ˆ M(G, z) = {µ | ∃p(x) ∈ ∆ s. [sent-78, score-0.315]

29 Furthermore, the µ that maximizes the LP on ˆ the right is integral and corresponds to the second-best MAP assignment x(2) . [sent-85, score-0.263]

30 Intuitively, it would seems that M(G, z) should be “similar” to M(G), such that it can be described as M(G) plus some constraints that “block” the assignment z. [sent-89, score-0.195]

31 Thus, this inequality separates ˆ v(z) from all other integral vertices. [sent-93, score-0.202]

32 The diﬃculty is that the resulting polytope has fractional vertices,3 and maximizing over it won’t generally yield an integral solution. [sent-95, score-0.477]

33 It turns out that there is a diﬀerent inequality that does yield an exact characterization of ˆ M(G, z) when G is a tree. [sent-96, score-0.171]

34 Consider the functional I(µ, z) (which is linear in µ): (1 − di )µi (zi ) + I(µ, z) = i µij (zi , zj ) (5) ij∈E where di is the degree of node i in the tree graph G. [sent-99, score-0.546]

35 4 The theorem also generalizes to the case where G is not a tree, but we have a junction tree for G. [sent-106, score-0.315]

36 In this case, the theorem still holds if we deﬁne a generalized I(µ, z) inequality as: (1 − dS )µS (zS ) + S∈S µC (zC ) ≤ 0 (7) C∈C where C and S are the junction tree cliques and their separators, respectively, and dS is the number of cliques that intersect on separator S. [sent-107, score-0.56]

37 In this case, the marginal polytope should enforce consistency between marginals µC (zC ) and their separators µS (zS ). [sent-108, score-0.447]

38 However, such a characterization requires variables whose cardinality is exponential in the tree-width and is thus tractable only for graphs of low tree-width. [sent-109, score-0.188]

39 3 Consider the case of a single edge between 2 nodes where the MAP assignment is (0, 0). [sent-115, score-0.136]

40 Adding the inequality µ1 (0) + µ2 (0) ≤ 1 produces the fractional vertex (0. [sent-116, score-0.221]

41 3 2nd best LPs for general graphs - Spanning tree inequalities 3 When the graph G is not a tree, the marginal polytope M(G) generally requires an exponential number of inequalities. [sent-121, score-0.855]

42 However, as mentioned above, it does have an exact description in terms of marginals over cliques and separators of a junction tree. [sent-122, score-0.345]

43 Given such marginals on ˆ junction tree cliques, we also have an exact characterization of M(G, z) via the constraint in Eq. [sent-123, score-0.519]

44 Thus a common strategy [15] is to replace M(G) with an outer bound that enforces consistency between marginals on overlapping sets of variables. [sent-126, score-0.134]

45 Moreover, direct application of the inequality will incorrectly remove some integral vertices. [sent-135, score-0.202]

46 An alternative approach is to add inequalities that separate v(z) from the other integral vertices. [sent-136, score-0.292]

47 This will serve to eliminate more and more fractional vertices, and if enough constraints are added, this may result in an integral solution. [sent-137, score-0.313]

48 One obvious family of such constraints are those corresponding to spanning trees in G and have the form of Eq. [sent-138, score-0.381]

49 We refer to I T (µ, z) ≤ 0 as a spanning tree inequality. [sent-143, score-0.497]

50 i For any sub-tree T of G, the corresponding spanning tree inequality separates the vertex v(z) from the other vertices. [sent-144, score-0.626]

51 Note, however, that the resulting polytope may still have fractional vertices. [sent-146, score-0.35]

52 The above argument shows that any spanning tree provides a separating inequality for ˆ M(G, z). [sent-147, score-0.572]

53 In principle, we would like to use as many such inequalities as possible. [sent-148, score-0.165]

54 The spanning tree assignment-excluding marginal polytope is deﬁned as: ˆ MST (G, z) = µ | µ ∈ ML (G), L ∀ tree T ⊆ E I T (µ, z) ≤ 0 (9) where the ST notation indicates the inclusion of all spanning tree inequalities for G. [sent-150, score-1.697]

55 , we seek the optimal µ subject to all spanning tree inequalities for G with the ambition that this µ be integral and thus provide the non-z MAP assignment, with a certiﬁcate of optimality. [sent-153, score-0.789]

56 Although the number of spanning trees is exponential in n, it turns out that all spanning inequalities can be used in practice. [sent-154, score-0.794]

57 One way to achieve this is via a cutting plane algorithm [12] that ﬁnds the most violated spanning tree inequality and adds it to the LP. [sent-155, score-0.695]

58 Then, as long as the optimal µ is fractional, we ﬁnd the spanning tree inequality that µ most violates (where this is implemented via the maximum-weight spanning tree). [sent-160, score-0.846]

59 If there are no violated inequalities 5 ˆ ˆL Note that M(G, z) ⊆ MST (G, z) ⊂ ML (G). [sent-162, score-0.204]

60 4 but µ is still fractional, then spanning tree inequalities do not suﬃce to ﬁnd an integral solution (but see below on hypertree constraints to add in this case). [sent-163, score-0.88]

61 In practice, we found that only a relatively small number of inequalities are needed to successfully yield an integral solution, or determine that all such inequalities are already satisﬁed. [sent-164, score-0.457]

62 The dual variables roughly correspond to points in the spanning tree polytope [16], optimization over which can be done in polynomial time, e. [sent-166, score-0.755]

63 A practical compromise would be to use inequalities over clique trees of G, where the cliques are relatively small, e. [sent-172, score-0.298]

64 7 with the small cliques and their separators) will necessarily separate v(z) from the other integral vertices. [sent-176, score-0.212]

65 Finding the maximally violated such inequality is an NP-hard problem, equivalent to a prize collecting Steiner tree problem, but recent work has found that such problems are often exactly solvable in practice [7]. [sent-177, score-0.337]

66 It thus might be practical to include all such trees as constraints using a cutting plane algorithm. [sent-178, score-0.191]

67 We begin by describing an algorithm for the exact M best and then show how it can be used to approximate those via the approximations for 2nd best described above. [sent-181, score-0.179]

68 It is inspired by several pre-existing M best solution schemes [4, 6, 8, 19] but diﬀers from them in highlighting the role of ﬁnding a second best solution within a given subspace. [sent-185, score-0.17]

69 5 S+R N+R B+R 0 S+R B B+R 0 S+R B B+R Figure 2: Number of best ranks and normalized run-times for the attractive and mixed grids, and the more diﬃcult protein SCP problems. [sent-195, score-0.192]

70 The correctness of PES can be shown by observing that at the M th stage, all previous best solutions are excluded from the optimization and no other assignment is excluded. [sent-199, score-0.263]

71 Of note, this simple partitioning scheme is possible due to the observation that the ﬁrst-best and second-best MAP assignments must diﬀer in the assignment of at least one variable in the graph. [sent-200, score-0.264]

72 We choose as ˆ this approximation the polytope MST (G, x(∗) ) in Eq. [sent-206, score-0.258]

73 Nilsson’s algorithm is equivalent to PES where the 2nd best assignment is obtained from maximizations within O(n) partitions, so that its runtime is O(n) times the cost of ﬁnding a single MAP. [sent-211, score-0.189]

74 Here we approximated each MAP with its LP relaxation (as in STRIPES), so that both STRIPES and Nilsson come with certiﬁcates of optimality when their LP solutions are integral. [sent-212, score-0.162]

75 Overall, the STRIPES algorithm was required to employ up to 19 spanning tree inequalities per calculation of second-best solution. [sent-228, score-0.662]

76 6 This is very diﬀerent from the second best constraint, since setting x1 = 1 blocks all assignments with this value, as opposed to setting x = 1 which blocks only the assignment with all ones. [sent-229, score-0.256]

77 STRIPES added up to 23 spanning tree inequalities per iteration. [sent-248, score-0.662]

78 The protein side-chain prediction (SCP) problem is to to predict the placement of amino acid side-chains given a protein backbone [2, 18]. [sent-249, score-0.305]

79 Minimization of a protein energy function corresponds to ﬁnding a MAP assignment for a pairwise MRF [19]. [sent-250, score-0.233]

80 For 315 of 370 problems in the dataset, the ﬁrst MAP solution was obtained directly as a result of the LP relaxation having an integral solution (“easy” problems). [sent-252, score-0.241]

81 5 hours for all but one of these cases (up to 8 spanning trees per calculation), and BMMF found the same 50 solutions for each case within 0. [sent-254, score-0.396]

82 Surprisingly, it was able to exactly ﬁnd the 50 top solutions for all cases, using up to 4 standard spanning tree inequalities per second-best calculation. [sent-260, score-0.736]

83 On the other hand, whether running BMMF without these regions (BP) or with the regions (GBP), it did not perform as well as STRIPES in terms of the number of high-ranking solutions or its speed. [sent-262, score-0.14]

84 To summarize, STRIPES provably found the top 50 solutions for 369 of the 370 protein SCP problems. [sent-263, score-0.224]

85 We provide a simple characterization of it for tree structured graphs, and show how it can be used for approximations in non-tree graphs. [sent-265, score-0.32]

86 Another compelling question is in which problems the spanning tree inequalities are provably optimal. [sent-270, score-0.715]

87 Here we studied the polytope that results from excluding one assignment. [sent-274, score-0.258]

88 An intriguing question is to characterize the polytope that excludes M assignments. [sent-275, score-0.258]

89 Finally, our empirical results show that our method indeed leverages the power of LP relaxations and yields exact M best optimal solutions for problems with large tree-width. [sent-285, score-0.268]

90 Denote G as the tree obtained ˆ by removing node n and its edge with n − 1. [sent-297, score-0.223]

91 For any assignment x, denote x as the corresponding sub-assignment for the ﬁrst n − 1 variables. [sent-298, score-0.136]

92 For an integral vertex µ = v(x), ˆˆ ˆ ˆ ˆ ˆ x denote its projected µ as v (ˆ ). [sent-300, score-0.181]

93 For ˆ any µ and its projected µ, it can be seen that: ˆˆ ˆ I(µ, z) = I(µ, z ) − α (11) where we deﬁne α = xn =zn µn−1,n (zn−1 , xn ) (so 0 ≤ α ≤ 1). [sent-303, score-0.154]

94 From the inductive assumption, pµ (ˆ ) = 0, so we deﬁne: ˆˆ z µn−1,n (xn−1 , xn ) p(x) = p(ˆ ) ˆx (12) µn−1 (xn−1 ) which indeed marginalizes to µ, and p(z) = 0 so that pµ (z) = 0 as required. [sent-308, score-0.195]

95 Note that this construction ˆx is identical to that used in proving that ML (G) = M(G) for a tree graph G. [sent-310, score-0.223]

96 Now, deﬁne p(x) so that p(z) = I(µ, z): ˆ assumption to µ, we obtain I( ˆ ˆ ˆˆ z xl , l ≤ n − 2 δ(xn−1 = zn−1 ) δ(xn = zn ) p(x) no constraint 0 no constraint As in Eq. [sent-316, score-0.179]

97 12 0 0 ∃ l x l = zl 1 ∀ l x l = zl 1 µn−1,n (zn−1 , xn ) 1 1 p(ˆ ) ˆx 0 I(µ, z) Simple algebra shows that p(x) is non-negative and has µ as marginals. [sent-317, score-0.211]

98 Deﬁne p(x) as: ˆ z ˆ xl , l ≤ n − 2 δ(xn−1 = zn−1 ) δ(xn = zn ) no constraint 0 no constraint ∃ l x l = zl As in Eq. [sent-330, score-0.246]

99 An algorithmic framework for the exact solution of the prize-collecting steiner tree problem. [sent-384, score-0.333]

100 Linear programming relaxations and belief propagation – an empirical study. [sent-465, score-0.137]

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Abstract: We propose a new model for natural image statistics. Instead of minimizing dependency between components of natural images, we maximize a simple form of dependency in the form of tree-dependencies. By learning ﬁlters and tree structures which are best suited for natural images we observe that the resulting ﬁlters are edge ﬁlters, similar to the famous ICA on natural images results. Calculating the likelihood of an image patch using our model requires estimating the squared output of pairs of ﬁlters connected in the tree. We observe that after learning, these pairs of ﬁlters are predominantly of similar orientations but different phases, so their joint energy resembles models of complex cells. 1 Introduction and related work Many models of natural image statistics have been proposed in recent years [1, 2, 3, 4]. 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This model has been applied to many scenarios, and proved to be one of the great successes of natural image statistics modeling with the emergence of edge-ﬁlters [5]. This approach has two problems. The ﬁrst is that dependencies between components are still very strong, even with those learned transformation seeking to remove them. Second, it has been shown that ICA achieves, after the learned transformation, only marginal gains when measured quantitatively against simpler method like PCA [7] in terms of redundancy reduction. A different approach was taken recently in the form of radial Gaussianization [8], in which components which are distributed in a radially symmetric manner are made independent by transforming them non-linearly into a radial Gaussian, and thus, independent from one another. A more elaborate approach, related to ICA, is Independent Subspace Component Analysis or ISA. 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Given these robust dependencies between ﬁlter outputs, it is somewhat peculiar that in order to get simple cell properties one needs to assume independence. In this work we ask whether it is possible to obtain V1 like ﬁlters in a model that assumes dependence. In our model we assume the ﬁlter distribution can be described by a tree graphical model [10] (see Figure 1). Degenerate cases of tree graphical models include ICA (in which no edges are present) and ISA (in which edges are only present within a subspace). But in its non-degenerate form, our model assumes any two ﬁlter outputs may be dependent. We allow components to belong to more than one subspace, and as a result, do not require independence between them. 2 Model and learning Our model is comprised of three main components. Given a set of patches, we look for the parameters which maximize the likelihood of a whitened natural image patch z: N p(yi |ypai ; β) p(z; W, β, T ) = p(y1 ) (3) i=1 Where y = Wz, T is the tree structure, pai denotes the parent of node i and β is a parameter of the density model (see below for the details). The three components we are trying to learn are: 1. The ﬁlter matrix W, where every row deﬁnes one of the ﬁlters. The response of these ﬁlters is assumed to be tree-dependent. We assume that W is orthogonal (and is a rotation of a whitening transform). 2. The tree structure T which speciﬁes which components are dependent on each other. 3. The probability density function for connected nodes in the tree, which specify the exact form of dependency between nodes. All three together describe a complete model for whitened natural image patches, allowing likelihood estimation and exact inference [11]. We perform the learning in an iterative manner: we start by learning the tree structure and density model from the entire data set, then, keeping the structure and density constant, we learn the ﬁlters via gradient ascent in mini-batches. Going back to the tree structure we repeat the process many times iteratively. It is important to note that both the ﬁlter set and tree structure are learned from the data, and are continuously updated during learning. In the following sections we will provide details on the speciﬁcs of each part of the model. 2 β=0.0 β=0.5 β=1.0 β=0.0 β=0.5 β=1.0 2 1 1 1 1 1 1 0 −1 0 −1 −2 −1 −2 −3 0 x1 2 0 x1 2 0 x1 2 −2 −3 −2 0 x1 2 0 −1 −2 −3 −2 0 −1 −2 −3 −2 0 −1 −2 −3 −2 0 x2 3 2 x2 3 2 x2 3 2 x2 3 2 x2 3 2 x2 3 −3 −2 0 x1 2 −2 0 x1 2 Figure 2: Shape of the conditional (Left three plots) and joint (Right three plots) density model in log scale for several values of β, from dependence to independence. 2.1 Learning tree structure In their seminal paper, Chow and Liu showed how to learn the optimal tree structure approximation for a multidimensional probability density function [12]. This algorithm is easy to apply to this scenario, and requires just a few simple steps. First, given the current estimate for the ﬁlter matrix W, we calculate the response of each of the ﬁlters with all the patches in the data set. Using these responses, we calculate the mutual information between each pair of ﬁlters (nodes) to obtain a fully connected weighted graph. The ﬁnal step is to ﬁnd a maximal spanning tree over this graph. The resulting unrooted tree is the optimal tree approximation of the joint distribution function over all nodes. We will note that the tree is unrooted, and the root can be chosen arbitrarily - this means that there is no node, or ﬁlter, that is more important than the others - the direction in the tree graph is arbitrary as long as it is chosen in a consistent way. 2.2 Joint probability density functions Gabor ﬁlter responses on natural images exhibit highly kurtotic marginal distributions, with heavy tails and sharp peaks [13, 3, 14]. Joint pair wise distributions also exhibit this same shape with varying degrees of dependency between the components [13, 2]. The density model we use allows us to capture both the highly kurtotic nature of the distributions, while still allowing varying degrees of dependence using a mixing variable. We use a mix of two forms of ﬁnite, zero mean Gaussian Scale Mixtures (GSM). In one, the components are assumed to be independent of each other and in the other, they are assumed to be spherically distributed. The mixing variable linearly interpolates between the two, allowing us to capture the whole range of dependencies: p(x1 , x2 ; β) = βpdep (x1 , x2 ) + (1 − β)pind (x1 , x2 ) (4) When β = 1 the two components are dependent (unless p is Gaussian), whereas when β = 0 the two components are independent. For the density functions themselves, we use a ﬁnite GSM. 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The mixing parameter β is also estimated using EM, but this is done for each edge in the tree separately, thus allowing our model to theoretically capture the fully independent case (ICA) and other degenerate models such as ISA. 2.3 Learning tree dependent components Given the current tree structure and density model, we can now learn the matrix W via gradient ascent on the log likelihood of the model. All learning is performed on whitened, dimensionally 3 reduced patches. This means that W is a N × N rotation (orthonormal) matrix, where N is the number of dimensions after dimensionality reduction (see details below). Given an image patch z we multiply it by W to get the response vector y: y = Wz (7) Now we can calculate the log likelihood of the given patch using the tree model (which we assume is constant at the moment): N log p(yi |ypai ) log p(y) = log p(yroot ) + (8) i=1 Where pai denotes the parent of node i. 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These four synthetic random data sets were given to the algorithm - results can be seen in Figure 3 for the ICA, ISA and tree samples. In all cases the model learned the ﬁlters and distribution correctly, reproducing both the ﬁlters (up to rotations within the subspace in ISA) and the dependency structure between the different ﬁlters. In the case of 1/f Gaussian noise, any whitening transformation is equally likely and any value of beta is equally likely. Thus in this case, the algorithm cannot ﬁnd the tree or the ﬁlters. 3.2 Learning from natural image patches We then ran experiments with a set of natural images [9]1 . These images contain natural scenes such as mountains, ﬁelds and lakes. . The data set was 50,000 patches, each 16 × 16 pixels large. The patches’ DC was removed and they were then whitened using PCA. Dimension was reduced from 256 to 128 dimensions. The GSM for the density model had 16 components. Several initial 1 available at http://www.cis.hut.ﬁ/projects/ica/imageica/ 4 Figure 3: Validation of the algorithm. Noise was generated from three models - top row is ICA, middle row is ISA and bottom row is a tree model. Samples were then given to the algorithm. On the right are the resulting learned tree models. Presented are the learned ﬁlters, tree model (with white edges meaning β = 0, black meaning β = 1 and grays intermediate values) and an example of a marginal histogram for one of the ﬁlters. It can be seen that in all cases all parts of the model were correctly learned. Filters in the ISA case were learned up to rotation within the subspace, and all ﬁlters were learned up to sign. β values for the ICA case were always below 0.1, as were the values of β between subspaces in ISA. conditions for the matrix W were tried out (random rotations, identity) but this had little effect on results. Mini-batches of 10 patches each were used for the gradient ascent - the gradient of 10 patches was summed, and then normalized to have unit norm. The learning rate constant η was set to 0.1. Tree structure learning and estimation of the mixing variable β were done every 500 mini-batches. All in all, 50 iterations were done over the data set. 3.3 Filters and tree structure Figures 4 and 5 show the learned ﬁlters (WQ where Q is the whitening matrix) and tree structure (T ) learned from natural images. Unlike the ISA toy data in ﬁgure 3, here a full tree was learned and β is approximately one for all edges. The GSM that was learned for the marginals was highly kurtotic. It can be seen that resulting ﬁlters are edge ﬁlters at varying scales, positions and orientations. This is similar to the result one gets when applying ICA to natural images [5, 15]. More interesting is Figure 4: Left: Filter set learned from 16 × 16 natural image patches. Filters are ordered by PCA eigenvalues, largest to smallest. Resulting ﬁlters are edge ﬁlters having different orientations, positions, frequencies and phases. Right: The “feature” set learned, that is, columns of the pseudo inverse of the ﬁlter set. 5 Figure 5: The learned tree graph structure and feature set. It can be seen that neighboring features on the graph have similar orientation, position and frequency. See Figure 4 for a better view of the feature details, and see text for full detail and analysis. Note that the ﬁgure is rotated CW. 6 Optimal Orientation Optimal Frequency 3.5 Optimal Phase 7 3 0.8 6 2.5 Optimal Position Y 0.9 6 5 5 0.7 0.6 3 Child 1.5 4 Child 2 Child Child 4 3 2 1 1 0.4 0.3 2 0.5 0.5 0.2 0 1 0.1 0 0 1 2 Parent 3 4 0 0 2 4 Parent 6 8 0 0 1 2 3 Parent 4 5 6 0 0.2 0.4 0.6 Parent 0.8 1 Figure 6: Correlation of optimal parameters in neighboring nodes in the tree graph. Orientation, frequency and position are highly correlated, while phase seems to be entirely uncorrelated. This property of correlation in frequency and orientation, while having no correlation in phase is related to the ubiquitous energy model of complex cells in V1. See text for further details. Figure 7: Left: Comparison of log likelihood values of our model with PCA, ICA and ISA. Our model gives the highest likelihood. Right: Samples taken at random from ICA, ISA and our model. Samples from our model appear to contain more long-range structure. the tree graph structure learned along with the ﬁlters which is shown in Figure 5. It can be seen that neighboring ﬁlters (nodes) in the tree tend to have similar position, frequency and orientation. Figure 6 shows the correlation of optimal frequency, orientation and position for neighboring ﬁlters in the tree - it is obvious that all three are highly correlated. Also apparent in this ﬁgure is the fact that the optimal phase for neighboring ﬁlters has no signiﬁcant correlation. It has been suggested that ﬁlters which have the same orientation, frequency and position with different phase can be related to complex cells in V1 [2, 16]. 3.4 Comparison to other models Since our model is a generalization of both ICA and ISA we use it to learn both models. In order to learn ICA we used the exact same data set, but the tree had no edges and was not learned from the data (alternatively, we could have just set β = 0). For ISA we used a forest architecture of 2 node trees, setting β = 1 for all edges (which means a spherical symmetric distribution), no tree structure was learned. Both models produce edge ﬁlters similar to what we learn (and to those in [5, 15, 6]). The ISA model produces neighboring nodes with similar frequency and orientation, but different phase, as was reported in [2]. We also compare to a simple PCA whitening transform, using the same whitening transform and marginals as in the ICA case, but setting W = I. We compare the likelihood each model gives for a test set of natural image patches, different from the one that was used in training. There were 50,000 patches in the test set, and we calculate the mean log likelihood over the entire set. The table in Figure 7 shows the result - as can be seen, our model performs better in likelihood terms than both ICA and ISA. Using a tree model, as opposed to more complex graphical models, allows for easy sampling from the model. Figure 7 shows 20 random samples taken from our tree model along with samples from the ICA and ISA models. Note the elongated structures (e.g. in the bottom left sample) in the samples from the tree model, and compare to patches sampled from the ICA and ISA models. 7 40 40 30 30 20 20 10 1 10 0.8 0.6 0.4 0 0.2 0 0 1 2 3 Orientation 4 0 0 2 4 6 Frequency 8 0 2 4 Phase Figure 8: Left: Interpretation of the model. Given a patch, the response of all edge ﬁlters is computed (“simple cells”), then at each edge, the corresponding nodes are squared and summed to produce the response of the “complex cell” this edge represents. Both the response of complex cells and simple cells is summed to produce the likelihood of the patch. Right: Response of a “complex cell” in our model to changing phase, frequency and orientation. Response in the y-axis is the sum of squares of the two ﬁlters in this “complex cell”. 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Abstract: Learning a measure of similarity between pairs of objects is a fundamental problem in machine learning. It stands in the core of classiﬁcation methods like kernel machines, and is particularly useful for applications like searching for images that are similar to a given image or ﬁnding videos that are relevant to a given video. In these tasks, users look for objects that are not only visually similar but also semantically related to a given object. Unfortunately, current approaches for learning similarity do not scale to large datasets, especially when imposing metric constraints on the learned similarity. We describe OASIS, a method for learning pairwise similarity that is fast and scales linearly with the number of objects and the number of non-zero features. Scalability is achieved through online learning of a bilinear model over sparse representations using a large margin criterion and an efﬁcient hinge loss cost. OASIS is accurate at a wide range of scales: on a standard benchmark with thousands of images, it is more precise than state-of-the-art methods, and faster by orders of magnitude. On 2.7 million images collected from the web, OASIS can be trained within 3 days on a single CPU. The nonmetric similarities learned by OASIS can be transformed into metric similarities, achieving higher precisions than similarities that are learned as metrics in the ﬁrst place. This suggests an approach for learning a metric from data that is larger by orders of magnitude than was handled before. 1

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