cvpr cvpr2013 cvpr2013-164 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Hilton Bristow, Anders Eriksson, Simon Lucey
Abstract: Sparse coding has become an increasingly popular method in learning and vision for a variety of classification, reconstruction and coding tasks. The canonical approach intrinsically assumes independence between observations during learning. For many natural signals however, sparse coding is applied to sub-elements (i.e. patches) of the signal, where such an assumption is invalid. Convolutional sparse coding explicitly models local interactions through the convolution operator, however the resulting optimization problem is considerably more complex than traditional sparse coding. In this paper, we draw upon ideas from signal processing and Augmented Lagrange Methods (ALMs) to produce a fast algorithm with globally optimal subproblems and super-linear convergence.
Reference: text
sentIndex sentText sentNum sentScore
1 au Abstract Sparse coding has become an increasingly popular method in learning and vision for a variety of classification, reconstruction and coding tasks. [sent-7, score-0.462]
2 For many natural signals however, sparse coding is applied to sub-elements (i. [sent-9, score-0.437]
3 Convolutional sparse coding explicitly models local interactions through the convolution operator, however the resulting optimization problem is considerably more complex than traditional sparse coding. [sent-12, score-0.619]
4 In this paper, we draw upon ideas from signal processing and Augmented Lagrange Methods (ALMs) to produce a fast algorithm with globally optimal subproblems and super-linear convergence. [sent-13, score-0.163]
5 nN=1||xn− Dzn||22+ β||zn||1 subject to | |dk | |22 ≤ 1 for k = 1. [sent-20, score-0.05]
6 K, (1) where β controls the L1 penalty, and the inequality constraint on the columns of D prevent the dictionary from absorbing all of the system’s energy. [sent-23, score-0.084]
7 A selection of filters learned from an unaligned set of lions. [sent-26, score-0.164]
8 The spatially invariant algorithm produces expression of generic Gabor-like filters as well as specialized domain specific filters, such as the highlighted “eye” . [sent-27, score-0.185]
9 Sparse coding has a fundamental drawback however, as it assumes the ensemble of input vectors are tin adsesupmenedsen tht eof e one abnleot ohfe irn, pi. [sent-28, score-0.231]
10 This independence assumption, when applied to nat- {xn}nN=1 ural images, leads to many basis elements that are translated versions of each other. [sent-31, score-0.046]
11 Convolutional sparse coding attempts to remedy this shortcoming by modelling shift invariance directly within the objective, argmdi,zn subject to 21||x −? [sent-32, score-0.45]
12 (2) Now zk takes the role of a sparse feature map which, when convolved with a filter dk and added over all k, should approximate the input signal x: a full signal 333898991 ( e. [sent-38, score-0.72]
13 image or audio sequence) rather than independent patches as in the objective of Equation (1) . [sent-40, score-0.13]
14 Like traditional sparse coding the estimated sparse basis {dk}kK=1 twioilnl able s pofa a efi cxoeddi nspga tthiael e support. [sent-41, score-0.541]
15 Hspoawrseeve bra, uisn {lidke} traditional sparse coding the input signal x and the sparse feature maps {zk}kK=1 are of a different and usually mfeautcuhr ela mrgearp sdim {zen}sionality. [sent-42, score-0.587]
16 Note also that we assume there is only a single signal x in our formulation in Equation (2) ; it is trivial in our proposed formulation to handle multiple signals each of varying length. [sent-43, score-0.166]
17 We persist with the single signal assumption throughout the derivation of our approach for the sake of clarity and brevity. [sent-44, score-0.092]
18 A 2D convolution operation is represented as the ∗ operatcoonr. [sent-52, score-0.124]
19 We have chosen to use a Fourier representation in this paper due to its particularly useful ability to represent convolutions as a Hadamard product in the Fourier domain. [sent-55, score-0.042]
20 represents tthhee H faacdtam thaatrd product, =anˆ zd diag() ish an operator tehnatts transforms a D dimensional vector into a D D ditmreannssifoonrmals diagonal menatsriioxn. [sent-58, score-0.119]
21 a Commutativity Dho ×lds D Dw dithithis property such that role of filter or signal ˆa can be interchanged. [sent-59, score-0.147]
22 Any transpose operator T on a complex vector or matrix in this paper additionally takes the complex conjugate in a similar fashion to the Hermitian adjoint [17] . [sent-60, score-0.095]
23 With respect to Equation (2) the objective of central interest in this paper – we will often omit filter indices (e. [sent-61, score-0.145]
24 dk refers to the kth filter and zk refers to the kth filter response) when referring z z to the variables being optimized. [sent-63, score-0.493]
25 z z Prior Art: The idea that sparse part-based representations form the computational foundations of visual systems has been supported by Olshausen and Field [15, 16] and many other studies [8, 20, 21] . [sent-75, score-0.165]
26 Neurons in the inferotemporal cortex respond to moderately complex features which are invariant to the position of stimulus within the visual field. [sent-76, score-0.109]
27 The resulting features were complex patterns rather than the Gabor-like features obtained by sparse coding. [sent-78, score-0.132]
28 The idea of truly shift-invariant or “convolutional” sparse coding was first proposed by Lewicki and Sejnowski for discrete 1D time-varying signals [12] , and later generalized to images by Mrup et al. [sent-80, score-0.437]
29 ’s work in convolutional sparse coding was motivated by the study of deconvolutional networks [26, 27] , which are closely related to the seminal works of Lecun on convolutional networks [9, 10] . [sent-83, score-1.027]
30 proposed to solve the objective in Equation (2) through an alternation strategy where one solves a sequence of convex subproblems until convergence. [sent-85, score-0.209]
31 The approach alternates between solving the subproblem d given a fixed z, and the subproblem z given a fixed d. [sent-86, score-0.586]
32 A drawback to this strategy however, is the computational overhead associated with both subproblems. [sent-87, score-0.048]
33 The introduction of convolution necessitates the use of gradient solvers for each subproblem, with linear convergence properties dramatically affecting the convergence properties of the overall algorithm. [sent-88, score-0.415]
34 Zeiler further introduced an auxiliary variable, t, to separate the convolution from the L1 regularization (allowing for an explicit and efficient solution to t using soft thresholding) . [sent-89, score-0.197]
35 Instead of enforcing the equality constraint z = t explicitly, the authors add a quadratic term 2μ | |z − t | |22 to penalize violations. [sent-90, score-0.108]
36 This quadratic penalty can |b|e reinterpreted as a trust region constraint | |z −t | |22 ≤ ? [sent-91, score-0.131]
37 In order to satisfy tshtrea equality constraint, μ m ? [sent-94, score-0.066]
38 The optimal value of subproblem d requires solution of a QCQP. [sent-97, score-0.293]
39 To avoid this added computational burden, Zeiler normalizes the solution to an unconstrained minimization, and while this tends to work in practice, it is an approximation not guaranteed to converge to the global minima of the original constrained objective (see Figure 2) . [sent-98, score-0.159]
40 Similar to Zeiler’s method is FISTA, from the family of proximal gradient methods [1] . [sent-99, score-0.038]
41 It is a well known iterative method capable of solving L1 regularized least squares problems with quadratic convergence properties. [sent-100, score-0.132]
42 m Aeuthgomde notfe dm uLlatgipralinergse ( mAeDthModMs) we use – – hs uavche similar quadratic convergence properties under more modest conditions [2, 25] and through their capacity 333999002 to compose functions, present fast, scalable and distributed solvers. [sent-103, score-0.17]
43 We argue that an algorithmic speedup can be obtained by applying an ADMM approach to the objective as a whole rather than the L1 subproblem alone. [sent-105, score-0.383]
44 • We demonstrate that the convolution subproblWeme can oben tsoralvteed heffaitci tehnetly c aonnvdo eluxtpiloicnit lsyu bipnr tohbeFourier domain; outperforming conventional gradient solvers that use spatial convolution. [sent-106, score-0.159]
45 By incorporating this approach within an ADMM optimization strategy the inequality constraints on • • the norm of the dictionary elements can be satisfied exactly by scaling the solution to an isotropic problem through the introduction of an additional auxillary variable. [sent-107, score-0.167]
46 We propose a quad-decomposition of the objectWivee pinrotop ssueb apr qoublaedm-dse ctohmatp are convex aen odb can be solved efficiently and explicitly without the need for gradient or sparse solvers. [sent-108, score-0.132]
47 As a result, we demonstrate an improvement in the computational efficiency of convolutional sparse coding over canonical methods (i. [sent-109, score-0.635]
48 Finally, we present a convolutional sparse coding alillbyr,ar wy,e w phreicshe can p cloungv-oalnudt-ipolnaayl d sipreacrtsely cinodtoexisting image and audio coding applications: hiltonbristow . [sent-113, score-0.906]
49 Problem Reformulation Our proposed approach to solving convolutional sparse coding involves the introduction of two auxiliary variables t and s as well as the posing of the convolutional portion of the objective in the Fourier domain, argd m,si,zn,t subject to 21D|| ˆx −k? [sent-116, score-1.154]
50 (3) Φ is a D M submatrix of the Fourier matrix F = [ΦΦ, i Φs ⊥a] Dth ×at Mcor sruebsmpoantdrsix xto o a shmea Fllo supriaetria ml support o=f the filter where M ? [sent-129, score-0.055]
51 Fourier convolution is not etxhaect fillyte req wuhivearelen Mt t ? [sent-131, score-0.124]
52 Ignoring these for the moment (see Boundary Effects on mitigating these differences) the objective in Equation (3) is equivalent to the original objective in Equation (2) . [sent-134, score-0.18]
53 In our proposed Fourier formulation is a D dimensional vector like ˆx and zk, whereas in the original spatial formulation in Equation (2) dk ∈ RM is of a significantly smaller dimensionality to M∈ ? [sent-135, score-0.142]
54 en Dforc ceo rtrheesmaller spatial constraint through the auxiliary variable s, which now becomes separable (in terms of variables) to the convolutional component of the objective. [sent-138, score-0.345]
55 original approach, we also separate the L1 penalty term from the convolutional component of the objective using the auxiliary variable t. [sent-140, score-0.524]
56 dˆk ’s Augmented Lagrangian: In this paper we propose to handle the introduced equality constraints through an augmented Lagrangian approach [3] . [sent-141, score-0.146]
57 The augmented Lagrangian of our proposed objective can be formed as, L(d, s, z, t, λs, λt) = 21D|| ˆx −k? [sent-142, score-0.17]
58 zˆ k||22+ β||t||1 + λsT(s − [ΦT ⊗ IK]dˆ) + λtT(z − t) +μ2s||s − [ΦT⊗ IK]dˆ||22 +μ2t||z − t||22 subject to | |sk | |22 ≤ 1 for k = 1. [sent-144, score-0.05]
59 (4) where λp and μp denote the Lagrange multiplier and penalty weighting for the two auxiliary variables p ∈ {pes,n ta}l yres wpeeigchtitvienlgy. [sent-148, score-0.288]
60 A full description of ADMMs is outside the scope of this paper (readers are encouraged to inspect [3] for a full treatment and review) , but they can be loosely interpreted as applying a GaussSeidel optimization strategy to the augmented Lagragian objective. [sent-151, score-0.128]
61 Such a strategy is advantageous as it often leads to extremely efficient subproblem decompo- × sitions. [sent-152, score-0.341]
62 We detail each of the subproblems following: Subproblem z: z∗ = arg mzin L(z;d,s,t,λs, λt) (5) = F−1{argm zˆin21|| xˆ −Dˆ zˆ||22+ ˆλtT(ˆ z −tˆ) +μ2t|| zˆ −tˆ||22} (6) = F−1{(DˆTDˆ + μtI)−1(DˆT xˆ + μtˆt −λˆt)} (7) where Dˆ = [diag(ˆd1) , . [sent-159, score-0.114]
63 Although the size of the matrix Dˆ is KD KD, it is sparse banded, and an efficient Dvar iisab KleD Dre ×or KdeDrin,g i tex i ssts sp (asersee Figure d3), such that the optimal z∗ can be found as the solution to D independent K K linear systems. [sent-163, score-0.132]
64 Subproblem t: t∗ = arg mtin L(t; d, s, z, λs, λt) (8) × argmtinμ2t||z − t||22+ λtT(z − t) + β||t||1 = (9) Unlike subproblem z, the solution to t cannot be efficiently computed in the Fourier domain, since the L1 norm is not rotation invariant. [sent-164, score-0.369]
65 Since the objective in Equation (9) does not contain any rotations of the data, each element of t = [t1, . [sent-166, score-0.09]
66 , tD]T can be solved independently, z λˆt = argmtinβ|t| + λ(z − t) +2μ(z − t)2 t∗ (10) where the optimal solution for each t can be found efficiently using the shrinkage function, t∗ = sgn? [sent-169, score-0.04]
67 Subproblem d: d∗ = arg msin L(d;s, z,t,λs, λt) = (12) F−1{argmdˆin21|| xˆ −Zˆdˆ||22+ λˆsT(ˆd − sˆ ) +μ2s||dˆ − sˆ ||22} = . [sent-173, score-0.093]
68 In a similar fashion to subproblem z, even though the size of the matrix is KD KD, a similar variable reordering exists suchZ t ish Kat Dfin×dKinDg ,t ahe s iomptiliamra val dria∗ inlev orelvoersd sroinlugti eoxn- Zˆ to D independent K Subproblem s: s∗ = = × K linear systems. [sent-178, score-0.366]
69 arg msin L(s;d,z,t,λs, λt) (15) argmsinμ2s||dˆ − [ΦT⊗ IK]s||22+ ˆλsT(dˆ − [ΦT ⊗ IK]s) subject to | |sk | |22 ≤ 1 for k = 1. [sent-179, score-0.143]
70 (16) In its general form, solving Equation (16) efficiently is problematic as it is a quadratically constrained quadratic programming (QCQP) problem. [sent-183, score-0.128]
71 Fortunately due to the kronecker product with the identity matrix IK it can be broken down into K independent problems, sk∗ = argmsiknμ2s||dˆk− ΦTsk||22+λˆsTk(dˆk− ΦTsk) subject to ||sk||22 ≤ 1 . [sent-184, score-0.05]
72 | ˜sk| ˜s|2k−,1 s˜k, oitfh | e ˜rswk|i 2 se≥ 1 where, (18) sk = (μsΦΦT)−1(Φdˆk + Φλˆsk) . [sent-186, score-0.189]
73 As a result one never needs to actually construct the sub-matrix Φ in order to estimate sk . [sent-194, score-0.189]
74 Lagrange Multiplier Update: λ(ti+1) λ(si+1) ← ← λt(i) + μt(z(i+1) − t(i+1)) λs(i) + μs(d(i+1) − s(i+1)) (21) (22) Penalty Update: Superlinear convergence of the ADMM may be achieved if μ(i) → ∞ [18] . [sent-195, score-0.09]
75 We use their “Fruit” dataset consisting of 10 images, apply local contrast normalization and select random 50 50 subimages Number of filters Number of input training images Figure 4. [sent-205, score-0.114]
76 Time to convergence as a function of (left) the number of filters learned with fixed number of input images, and (right) the number input images with fixed number of filters. [sent-206, score-0.204]
77 Comparison of objective when learning 64 filters from the experiments of Figure 4. [sent-208, score-0.204]
78 Our objective starts at a much larger value than Zeiler et al. [sent-209, score-0.09]
79 ’s, due to added Lagrange multipliers and penalty terms, but quickly converges to a good solution. [sent-210, score-0.199]
80 We perform two experiments, first holding the number of training examples fixed whilst varying the number of filters learned, then visa versa. [sent-212, score-0.114]
81 A representative contains Gabor-like set of 450 filters learned on a laptop using a collection of whitened components, as well as more expressive the spatially-invariant centre-surround natural images. [sent-216, score-0.114]
82 and cross-like components The set which appear since learning strategy produces less spatially shifted versions of the filters. [sent-217, score-0.083]
83 ’s method (red line) with respect to the number of filters being learned. [sent-219, score-0.114]
84 This is largely due to two contributing factors: (i) the direct methods we use to solve each subproblem have super-linear convergence properties, whereas the conjugate gradient method employed by Zeiler et al. [sent-220, score-0.432]
85 is limited to linear convergence, and (ii) convolution in the Fourier domain involves a simple Hadamard product whereas convolutions must be explicitly recomputed for each iteration of conjugate gradients. [sent-221, score-0.248]
86 Our method starts at a much larger objective value, due to the additional Lagrange multiplier and penalty terms. [sent-227, score-0.238]
87 The objective quickly decreases however, as these terms vanish and the equality constraints are satisfied. [sent-228, score-0.156]
88 The overall curve of our objective is typical of ADMMs: steep convergence to begin with, followed by flat-lining and minimal convergence beyond that point. [sent-229, score-0.27]
89 Applying sparse coding algorithms to the original Olshausen and Field dataset [15] has become a standard “sanity check” , to ensure that the method is capable of producing Gabor-like oriented edge filters. [sent-230, score-0.363]
90 While our convolutional algorithm produces some Gabor-like responses, it also has a greater expression of non-Gabor filters which are tailored more towards the semantics of the dataset. [sent-234, score-0.386]
91 Figure 1 shows a compelling example of this artifact, with one of the filters clearly synthesizing an “eye” feature from a set of unaligned lions. [sent-235, score-0.164]
92 Conclusions We presented a method for solving convolutional sparse coding problems in a fast manner through quaddecomposition of the original objective into subproblems that have an efficient parameterization in the Fourier domain. [sent-237, score-0.796]
93 These components working in union allow us to solve the rotation invariant L1 subproblem for each index independently using soft thresholding, and transform the quadratically constrained filter update equation to an unconstrained isotropic system. [sent-238, score-0.573]
94 As filter support size increases, the appeal of Fourier convolution becomes more apparent, and where boundary effects are problematic the Fourier transform can be seamlessly replaced by the Discrete Cosine Transform. [sent-239, score-0.213]
95 Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information. [sent-260, score-0.092]
96 Properties of basis functions generated by shift invariant sparse representations of natural images. [sent-288, score-0.253]
97 Shift invariant sparse coding of image and music data. [sent-329, score-0.401]
98 Emergence of simplecell receptive field properties by learning a sparse code for natural images. [sent-334, score-0.17]
99 Recognition of objects and their component parts: responses of single units in the temporal cortex of the macaque. [sent-369, score-0.071]
100 Linear spatial pyramid matching using sparse coding for image classification. [sent-386, score-0.363]
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