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145 nips-2007-On Sparsity and Overcompleteness in Image Models

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Author: Pietro Berkes, Richard Turner, Maneesh Sahani

Abstract: Computational models of visual cortex, and in particular those based on sparse coding, have enjoyed much recent attention. Despite this currency, the question of how sparse or how over-complete a sparse representation should be, has gone without principled answer. Here, we use Bayesian model-selection methods to address these questions for a sparse-coding model based on a Student-t prior. Having validated our methods on toy data, we ﬁnd that natural images are indeed best modelled by extremely sparse distributions; although for the Student-t prior, the associated optimal basis size is only modestly over-complete. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Despite this currency, the question of how sparse or how over-complete a sparse representation should be, has gone without principled answer. [sent-2, score-0.362]

2 Having validated our methods on toy data, we ﬁnd that natural images are indeed best modelled by extremely sparse distributions; although for the Student-t prior, the associated optimal basis size is only modestly over-complete. [sent-4, score-0.553]

3 1 Introduction Computational models of visual cortex, and in particular those based on sparse coding, have recently enjoyed much attention. [sent-5, score-0.285]

4 The basic assumption behind sparse coding is that natural scenes are composed of structural primitives (edges or lines, for example) and, although there are a potentially large number of these primitives, typically only a few are active in a single natural scene (hence the term sparse, [1, 2]). [sent-6, score-1.305]

5 The claim is that cortical processing uses these statistical regularities to shape a representation of natural scenes, and in particular converts the pixel-based representation at the retina to a higher-level representation in terms of these structural primitives. [sent-7, score-0.241]

6 Traditionally, research has focused on determining the characteristics of the structural primitives and comparing their representational properties with those of V1. [sent-8, score-0.482]

7 The two we focus on here are: How large is the set of structural primitives best suited to describe all natural scenes (how over-complete), and how many primitives are active in a single scene (how sparse)? [sent-10, score-1.159]

8 We will also be interested in the coupling between sparseness and over-completeness. [sent-11, score-0.4]

9 The intuition is that, if there are a great number of structural primitives, they can be very speciﬁc and only a small number will be active in a visual scene. [sent-12, score-0.204]

10 We attempt to map this coupling by evaluating models with different over-completenesses and sparsenesses and discover where natural scenes live along this trade-off (see Fig. [sent-14, score-0.504]

11 In order to test the sparse coding hypothesis it is necessary to build algorithms that both learn the primitives and decompose natural scenes in terms of them. [sent-16, score-0.944]

12 There have been many ways to derive such algorithms, but one of the more successful is to regard the task of building a representation of natural scenes as one of probabilistic inference. [sent-17, score-0.287]

13 More speciﬁcally, the unknown activities of the structural primitives are viewed as latent variables that must be inferred from the natural scene data. [sent-18, score-0.786]

14 Commonly the inference is carried out by writing down a generative model (although see [3] for an alternative), which formalises the assumptions made about the data and latent variables. [sent-19, score-0.196]

15 Unfortunately the assumption that natural scenes are composed of a small number of structural primitives is not sufﬁcient to build a meaningful generative model. [sent-21, score-0.82]

16 Other assumptions must therefore be made and typically these are that the primitives occur independently, and combine linearly. [sent-22, score-0.321]

17 These 1 10 overcompleteness 8 6 4 2 0 sparsity Figure 1: Schematic showing the space of possible sparse coding models in terms of sparseness (increasing in the direction of the arrow) and over-completeness. [sent-23, score-0.966]

18 via their marginal likelihood or cross-validation) and the grey contours illustrate what we might expect to discover if this were possible: The solid black line illustrates the hypothesised trade-off between over-completeness and sparsity, whilst the star shows the optimal point in this trade-off. [sent-27, score-0.261]

19 are drastic approximations and it is an open question to what extent this affects the results of sparse coding. [sent-28, score-0.181]

20 The distribution over the latent variables xt,k is chosen to be sparse and typical choices are Student-t, a Mixture of Gaussians (with zero means), and the Generalised Gaussian (which includes the Laplace distribution). [sent-29, score-0.275]

21 (2) The goal of this paper will be to learn the optimal dimensionality of the latent variables (K) and the optimal sparseness of the prior (α). [sent-31, score-0.644]

22 In fact, if the hypothesis is that the visual system is implementing an optimal generative model, then questions of over-completeness and sparsity should be addressed in this context. [sent-37, score-0.303]

23 Finally, we present results concerning the optimal sparseness and over-completeness for natural image patches in the case that the prior is a Student-t distribution. [sent-40, score-0.725]

24 Here, we focus on the Student-t prior for the latent variables xt,k : Γ α+1 p(xt,k |α, λ) = √ 2 λ απ Γ α 2 1 1+ α xt,k λ 2 − α+1 2 (3) There are two main reasons for this choice: The ﬁrst is that this is a widely used model [1]. [sent-42, score-0.203]

25 The second is that by implementing the Student-t prior using an auxiliary variable, all the distributions in the generative model become members of the exponential family [5]. [sent-43, score-0.171]

26 This means it is easy to derive efﬁcient approximate inference schemes like variational Bayes and Gibbs sampling. [sent-44, score-0.133]

27 These nonzero elements form star-like patterns, where the points of the star are determined by the direction of the weights (e. [sent-48, score-0.129]

28 One of the major technical difﬁculties posed by sparse-coding is that, in the over-complete regime, the posterior distribution of the latent variables p(X|Y, θ) is often complex and multi-modal. [sent-52, score-0.144]

29 Approximation schemes are therefore required, but we must be careful to ensure that the scheme we choose does not bias the conclusions we are trying to draw. [sent-53, score-0.137]

30 This is true for any application of sparse coding, but is particularly pertinent for our problem as we will be quantitatively comparing different sparse-coding models. [sent-54, score-0.272]

31 However, an alternative is to learn the optimal overcompleteness at a given sparseness using automatic relevance determination (ARD) [7, 8]. [sent-67, score-0.662]

32 In a nutshell, the idea is to equip the model with many more components than are believed to be present in the data, and to let it prune out the weights which are unnecessary. [sent-71, score-0.15]

33 Practically this involves placing a (Gaussian) prior over the components which favours small weights, and then inferring the scale of this prior. [sent-72, score-0.176]

34 4 (13) Determining the over-completeness: Variational Bayes In the previous two sections we described a generative model for sparse coding that is theoretically able to learn the optimal over-completeness of natural scenes. [sent-75, score-0.621]

35 We have two distinct uses for this model: The ﬁrst, and computationally more demanding task, is to learn the over-completeness at a variety of different, ﬁxed, sparsenesses (that is, to ﬁnd the optimal over-completeness in a vertical slice through Fig. [sent-76, score-0.148]

36 1); The second is to determine the optimal point on this trade-off by evaluating the (approximate) marginal-likelihood (that is, evaluating points along the trade-off line in Fig. [sent-77, score-0.148]

37 The ﬁrst approximation scheme is Variational Bayes (VB), and it excels at the ﬁrst task, but is severely biased in the case of the second. [sent-80, score-0.109]

38 In particular if we assume that the set of parameters and set of latent variables are independent in the posterior, so that q(V, Θ) = q(V )q(Θ) then we can sequentially optimise the free-energy with respect to each of these distributions. [sent-88, score-0.139]

39 The difference between the free energy and the true likelihood is given by the KL divergence between the approximate and true posterior. [sent-95, score-0.278]

40 At lowsparsities the prior becomes Gaussian-like and the posterior also becomes a uni-modal Gaussian. [sent-99, score-0.119]

41 One might also be concerned with another potential source of bias: The number of modes in the posterior increases with the number of components in the model, which gives a worse match to the variational approximation for more over-complete models. [sent-102, score-0.218]

42 However, because of the sparseness of the prior distribution, most of the modes are going to be very shallow for typical inputs, so that this effect should be small. [sent-103, score-0.406]

43 One alternative might be to use a variational method which has a multi-modal approximating distribution (e. [sent-109, score-0.128]

44 Brieﬂy, this inner loop starts by drawing samples from the model’s prior distribution and continues to sample as the prior is deformed into the posterior, according to an annealing schedule. [sent-115, score-0.261]

45 6 Results Before tackling natural images, it is necessary to verify that the approximations can discover the correct degree of over-completeness and sparsity in the case where the data are drawn from the forward model. [sent-118, score-0.36]

46 1 Veriﬁcation using simple artiﬁcal data In the ﬁrst experiment the training data are produced as follows: Two-dimensional observations are generated by three Student-t sources with degree of freedom chosen to be 2. [sent-121, score-0.116]

47 The generative weights are ﬁxed to be 60 degrees apart from one another, as shown in Figure 2. [sent-123, score-0.132]

48 A series of VB simulations were then run, differing only in the sparseness level (as measured by the degrees of freedom of the Student-t distribution over xt ). [sent-124, score-0.56]

49 Each simulation consisted of 500 VB iterations performed on a set of 3000 data points randomly generated from the model. [sent-125, score-0.148]

50 We initialised the simulations with K = 7 components. [sent-126, score-0.182]

51 To improve convergence, we started the simulations with weights near the origin (drawn from a normal distribution with mean 0 and standard deviation 10−8 ) and a relatively large input noise variance, and annealed the noise variance between the iterations 2 of VBEM. [sent-127, score-0.438]

52 The annealing schedule was as following: we started with σy = 0. [sent-128, score-0.269]

53 During the annealing process, the weights typically grew from the origin and spread in all directions to cover the input space. [sent-132, score-0.311]

54 At the same time, the corresponding precision hyperparameters grew and effectively pruned the unnecessary components. [sent-134, score-0.171]

55 We performed 7 blocks of simulations at different sparseness levels. [sent-135, score-0.462]

56 In every block we performed 3 runs of the algorithm and retained the result with the highest free energy. [sent-136, score-0.241]

57 5 −8000 −6500 −8200 −6550 log P(Y) (NATS) free energy (NATS) 4 −8400 −8600 −8800 −9000 −9200 2 0 2 −9400 −6600 −6650 −6700 −6750 −6800 −9600 4 2. [sent-137, score-0.24]

58 We derived the importance weights using a ﬁxed data set with 2500 data points, 250 samples, and 300 intermediate distributions. [sent-154, score-0.14]

59 Following the recommendations in [9], the annealing schedule was chosen to be linear initially (with 50 inverse temperatures spaced uniformly from 0 to 0. [sent-155, score-0.302]

60 The results indicate that the combination of the two methods is successful at learning both the overcompleteness and sparseness. [sent-159, score-0.235]

61 In particular the VBEM algorithm was able to recover the correct dimensionality for all sparseness levels, except for the sparsest case α = 2. [sent-160, score-0.337]

62 As expected, however, ﬁgure 2 shows that the maximum free energy is biased toward the more Gaussian models. [sent-162, score-0.297]

63 2), which is strictly greater than the free-energy as expected, favours sparseness levels close to the true value. [sent-164, score-0.52]

64 2 Veriﬁcation using complex artiﬁcial data Although it is necessary that the inference scheme should pass simple tests like that in the previous section, they are not sufﬁcient to give us conﬁdence that it will perform successfully on natural data. [sent-166, score-0.181]

65 One pertinent criticism is that the regime in which we tested the algorithms in the previous section (two dimensional observations, and three hidden latents) is quite different from that required to model natural data. [sent-167, score-0.325]

66 To that end, in this section we ﬁrst learn a sparse model for natural images with ﬁxed over-completeness levels using a Maximum A Posteriori (MAP) algorithm [2] (degree of freedom 2. [sent-168, score-0.607]

67 The goal is to validate the model on data which has a content and scale similar to the natural images case, but with a controlled number of generative components. [sent-171, score-0.308]

68 The image data comprised patches of size 9 × 9 pixels, taken at random positions from 36 natural images randomly selected from the van Hateren database (preprocessed as described in [10]). [sent-172, score-0.389]

69 The MAP solution was trained for 500 iterations, with every iteration performed on a new batch of 1440 patches (100 patches per image). [sent-174, score-0.244]

70 The model was initialised with a 3-times over-complete number of components (K = 108). [sent-175, score-0.187]

71 As above, the weights were initialised near the origin, and the input noise was annealed linearly from σd = 0. [sent-176, score-0.301]

72 Every run consisted of 500 VBEM iterations, with every iteration performed on 3600 patches generated from the MAP solution. [sent-179, score-0.184]

73 We performed several simulations for over-completeness levels between 0. [sent-180, score-0.203]

74 5, and retained the solutions with the highest free energy. [sent-182, score-0.191]

75 5 True Overcompleteness Figure 3: True versus inferred over-completeness from data drawn from the forward model trained on natural images. [sent-195, score-0.26]

76 This straight line saturates when we hit the number of latent variables with which ARD was initialised (three times overcomplete). [sent-197, score-0.251]

77 The results using multiple runs of ARD are close to this line (open circles, simulations with the highest free-energy are shown as closed circles). [sent-198, score-0.115]

78 The maximal and best over-completeness inferred from natural scenes is shown by the dotted line, and lies well below the over-completenesses we are able to infer. [sent-199, score-0.34]

79 values are still far above the highest over-completeness learned from natural images (see section 6. [sent-200, score-0.246]

80 3 Natural images Having established that the model performs as expected, at least when the data is drawn from the forward model, we now turn to natural image data and examine the optimal over-completeness ratio and sparseness degree for natural scene statistics. [sent-203, score-0.97]

81 The image data for this simulation and the model initialisation and annealing procedure are identical to the ones in the experiments in the preceeding section. [sent-204, score-0.202]

82 We performed 20 simulations with different sparseness levels, especially concentrated on the more sparse values. [sent-205, score-0.643]

83 As shown in Figure 4, the free energy increased almost monotonically until α = 5 and then stabilised and started to decrease for more Gaussian models. [sent-207, score-0.259]

84 3, with a trend for being more over-complete at high sparseness levels (Fig. [sent-209, score-0.46]

85 Although this general trend accords with the intuition that sparseness and over-completeness are coupled, both the magnitude of the effect and the degree of over-completeness is smaller than might have been anticipated. [sent-211, score-0.469]

86 Finally we performed AIS using the same annealing schedule as in Section 6. [sent-213, score-0.262]

87 1, using 250 samples for the ﬁrst 6 sparseness levels and 50 for the successive 14. [sent-214, score-0.415]

88 The estimates obtained for the log marginal likelihood, shown in Figure 4, were monotonically increasing with increasing sparseness (decreasing α). [sent-215, score-0.437]

89 This indicates that sparse models are indeed optimal for natural scenes. [sent-216, score-0.409]

90 Note that this is exactly the opposite trend to that of the free energy, indicating that it is also biased for natural scenes. [sent-217, score-0.339]

91 The weights resemble the Gabor wavelets, typical of sparse codes for natural images [1]. [sent-220, score-0.457]

92 7 Discussion Our results suggest that the optimal sparse-coding model for natural scenes is indeed one which is very sparse, but only modestly over-complete. [sent-221, score-0.493]

93 The anticipated coupling between the degree of sparsity and the over-completeness in the model is visible, but is weak. [sent-222, score-0.247]

94 One crucial question is how far these results will generalise to other prior distributions; and indeed, which of the various possible sparse-coding priors is best able to capture the structure of natural scenes. [sent-223, score-0.198]

95 a) Free energy b) Marginal likelihood c) Estimated over-completeness d) Basis vectors freedom parameter moves towards sparser values. [sent-239, score-0.16]

96 It is not clear that data with such extreme values will be encountered in typical data sets, and so the model may become distorted at high sparseness values. [sent-241, score-0.377]

97 Emergence of simple-cell receptive ﬁeld properties by learning a sparse code for natural images. [sent-252, score-0.31]

98 Sparse coding with an overcomplete basis set: A strategy employed by V1? [sent-259, score-0.186]

99 The ‘independent components’ of natural scenes are edge ﬁlters. [sent-274, score-0.287]

100 Independent component ﬁlters of natural images compared with simple cells in primary visual cortex. [sent-308, score-0.257]

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1 0.98053312 114 nips-2007-Learning and using relational theories

Author: Charles Kemp, Noah Goodman, Joshua B. Tenenbaum

Abstract: Much of human knowledge is organized into sophisticated systems that are often called intuitive theories. We propose that intuitive theories are mentally represented in a logical language, and that the subjective complexity of a theory is determined by the length of its representation in this language. This complexity measure helps to explain how theories are learned from relational data, and how they support inductive inferences about unobserved relations. We describe two experiments that test our approach, and show that it provides a better account of human learning and reasoning than an approach developed by Goodman [1]. What is a theory, and what makes one theory better than another? Questions like these are of obvious interest to philosophers of science but are also discussed by psychologists, who have argued that everyday knowledge is organized into rich and complex systems that are similar in many respects to scientiﬁc theories. Even young children, for instance, have systematic beliefs about domains including folk physics, folk biology, and folk psychology [2]. Intuitive theories like these play many of the same roles as scientiﬁc theories: in particular, both kinds of theories are used to explain and encode observations of the world, and to predict future observations. This paper explores the nature, use and acquisition of simple theories. Consider, for instance, an anthropologist who has just begun to study the social structure of a remote tribe, and observes that certain words are used to indicate relationships between selected pairs of individuals. Suppose that term T1(·, ·) can be glossed as ancestor(·, ·), and that T2(·, ·) can be glossed as friend(·, ·). The anthropologist might discover that the ﬁrst term is transitive, and that the second term is symmetric with a few exceptions. Suppose that term T3(·, ·) can be glossed as defers to(·, ·), and that the tribe divides into two castes such that members of the second caste defer to members of the ﬁrst caste. In this case the anthropologist might discover two latent concepts (caste 1(·) and caste 2(·)) along with the relationship between these concepts. As these examples suggest, a theory can be deﬁned as a system of laws and concepts that specify the relationships between the elements in some domain [2]. We will consider how these theories are learned, how they are used to encode relational data, and how they support predictions about unobserved relations. Our approach to all three problems relies on the notion of subjective complexity. We propose that theory learners prefer simple theories, that people remember relational data in terms of the simplest underlying theory, and that people extend a partially observed data set according to the simplest theory that is consistent with their observations. There is no guarantee that a single measure of subjective complexity can do all of the work that we require [3]. This paper, however, explores the strong hypothesis that a single measure will sufﬁce. Our formal treatment of subjective complexity begins with the question of how theories are mentally represented. We suggest that theories are represented in some logical language, and propose a speciﬁc ﬁrst-order language that serves as a hypothesis about the “language of thought.” We then pursue the idea that the subjective complexity of a theory corresponds to the length of its representation in this language. Our approach therefore builds on the work of Feldman [4], and is related to other psychological applications of the notion of Kolmogorov complexity [5]. The complexity measure we describe can be used to deﬁne a probability distribution over a space of theories, and we develop a model of theory acquisition by using this distribution as the prior for a Bayesian learner. We also 1 (a) Star 11 (b) Bipartite (c) Exception 22 33 44 55 66 77 88 16 26 36 46 56 21 31 41 51 61 71 81 11 17 27 37 47 57 18 28 38 48 58 R(X, X). T(6). T(7). T(8). R(X, Y) ← ¯(X), T(Y). T R(X, 1). (d) Symmetric (e) Transitive 11 22 33 44 55 66 77 13 31 12 21 24 42 56 65 26 36 46 56 17 27 37 47 57 18 28 38 48 58 T(6). T(7). T(8). R(X, Y) ← ¯(X), T(Y). T ¯ R(1, 1). R(1, 6). (f) Random 12 21 13 23 14 24 34 13 32 14 24 34 15 25 35 45 16 26 36 46 56 51 52 35 54 61 26 63 46 56 R(1, 2). R(1, 3). R(2, 4). R(5, 6). R(1, 2). R(2, 3). R(3, 4). R(5, X). R(X, 4). R(X, Y) ← R(Y, X). R(X, X). R(4, 5). R(5, 6). R(X, Z) ← R(X, Y), R(Y, Z). R(2, 1). R(1, 3). R(6, 1). R(3, 2). R(2, 6). R(3, 5). R(6, 3). R(4, 6). ¯ ¯ ¯ R(X, X). R(6, 4). R(5, 3). Figure 1: Six possible extensions for a binary predicate R(·, ·). In each case, the objects in the domain are represented as digits, and a pair such as 16 indicates that R(1, 6) is true. Below each set of pairs, the simplest theory according to our complexity measure is shown. show how the same Bayesian approach helps to explain how theories support inductive generalization: given a set of observations, future observations (e.g. whether one individual defers to another) can be predicted using the posterior distribution over the space of theories. We test our approach by developing two experiments where people learn and make predictions about binary and ternary relations. As far as we know, the approach of Goodman [1] is the only other measure of theory complexity that has previously been tested as a psychological model [6]. We show that our experiments support our approach and raise challenges for this alternative model. 1 Theory complexity: a representation length approach Intuitive theories correspond to mental representations of some sort, and our ﬁrst task is to characterize the elements used to build these representations. We explore the idea that a theory is a system of statements in a logical language, and six examples are shown in Fig. 1. The theory in Fig. 1b is related to the defers to(·, ·) example already described. Here we are interested in a domain including 9 elements, and a two place predicate R(·, ·) that is true of all and only the 15 pairs shown. R is deﬁned using a unary predicate T which is true of only three elements: 6, 7, and 8. The theory includes a clause which states that R(X, Y) is true for all pairs XY such that T(X) is false and T(Y) is true. The theory in Fig. 1c is very similar, but includes an additional clause which speciﬁes that R(1, 1) is true, and an exception which speciﬁes that R(1, 6) is false. Formally, each theory we consider is a collection of function-free deﬁnite clauses. All variables are universally quantiﬁed: for instance, the clause R(X, Z) ← R(X, Y), R(Y, Z) is equivalent to the logical formula ∀x ∀y ∀z (R(x, z) ← R(x, y) ∧ R(y, z)). For readability, the theories in Fig. 1 include parentheses and arrows, but note that these symbols are unnecessary and can be removed. Our proposed language includes only predicate symbols, variable symbols, constant symbols, and a period that indicates when one clause ﬁnishes and another begins. Each theory in Fig. 1 speciﬁes the extension of one or more predicates. The extension of predicate P is deﬁned in terms of predicate P+ (which captures the basic rules that lead to membership in P) and predicate P− (which captures exceptions to these rules). The resulting extension of P is deﬁned 2 as P+ \ P− , or the set difference of P+ and P− .1 Once P has been deﬁned, later clauses in the theory may refer to P or its negation ¯. To ensure that our semantics is well-deﬁned, the predicates P in any valid theory must permit an ordering so that the deﬁnition of any predicate does not refer to predicates that follow it in the order. Formally, the deﬁnition of each predicate P+ or P− can refer only to itself (recursive deﬁnitions are allowed) and to any predicate M or ¯ where M < P. M Once we have committed to a speciﬁc language, the subjective complexity of a theory is assumed to correspond to the number of symbols in its representation. We have chosen a language where there is one symbol for each position in a theory where a predicate, variable or constant appears, and one symbol to indicate when each clause ends. Given this language, the subjective complexity c(T ) of theory T is equal to the sum of the number of clauses in the theory and the number of positions in the theory where a predicate, variable or constant appears: c(T ) = #clauses(T ) + #pred slots(T ) + #var slots(T ) + #const slots(T ). (1) For instance, the clause R(X, Z) ← R(X, Y), R(Y, Z). contributes ten symbols towards the complexity of a theory (three predicate symbols, six variable symbols, and one period). Other languages might be considered: for instance, we could use a language which uses ﬁve symbols (e.g. ﬁve bits) to represent each predicate, variable and constant, and one symbol (e.g. one bit) to indicate the end of a clause. Our approach to subjective complexity depends critically on the representation language, but once a language has been chosen the complexity measure is uniquely speciﬁed. Although our approach is closely related to the notion of Kolmogorov complexity and to Minimum Message Length (MML) and Minimum Description Length (MDL) approaches, we refer to it as a Representation Length (RL) approach. A RL approach includes a commitment to a speciﬁc language that is proposed as a psychological hypothesis, but these other approaches aspire towards results that do not depend on the language chosen.2 It is sometimes suggested that the notion of Kolmogorov complexity provides a more suitable framework for psychological research than the RL approach, precisely because it allows for results that do not depend on a speciﬁc description language [8]. We subscribe to the opposite view. Mental representations presumably rely on some particular language, and identifying this language is a central challenge for psychological research. The language we described should be considered as a tentative approximation of the language of thought. Other languages can and should be explored, but our language has several appealing properties. Feldman [4] has argued that deﬁnite clauses are psychologically natural, and working with these representations allows our approach to account for several classic results from the concept learning literature. For instance, our language leads to the prediction that conjunctive concepts are easier to learn than disjunctive concepts [9].3 Working with deﬁnite clauses also ensures that each of our theories has a unique minimal model, which means that the extension of a theory can be deﬁned in a particularly simple way. Finally, human learners deal gracefully with noise and exceptions, and our language provides a simple way to handle exceptions. Any concrete proposal about the language of thought should make predictions about memory, learning and reasoning. Suppose that data set D lists the extensions of one or more predicates, and that a theory is a “candidate theory” for D if it correctly deﬁnes the extensions of all predicates in D. Note that a candidate theory may well include latent predicates—predicates that do not appear in D, but are useful for deﬁning the predicates that have been observed. We will assume that humans encode D in terms of the simplest candidate theory for D, and that the difﬁculty of memorizing D is determined by the subjective complexity of this theory. Our approach can and should be tested against classic results from the memory literature. Unlike some other approaches to complexity [10], for instance, our model predicts that a sequence of k items is about equally easy to remember regardless of whether the items are drawn from a set of size 2, a set of size 10, or a set of size 1000 [11]. 1 The extension of P+ is the smallest set that satisﬁes all of the clauses that deﬁne P+ , and the extension of P is deﬁned similarly. To simplify our notation, Fig. 1 uses P to refer to both P and P+ , and ¯ to refer to ¯ and P P P− . Any instance of P that appears in a clause deﬁning P is really an instance of P+ , and any instance of ¯ that P appears in a clause deﬁning ¯ is really an instance of P− . P 2 MDL approaches also commit to a speciﬁc language, but this language is often intended to be as general as possible. See, for instance, the discussion of universal codes in Gr¨ nwald et al. [7]. u 3 A conjunctive concept C(·) can be deﬁned using a single clause: C(X) ← A(X), B(X). The shortest deﬁnition of a disjunctive concept requires two clauses: D(X) ← A(X). D(X) ← B(X). − 3 To develop a model of inductive learning and reasoning, we take a Bayesian approach, and use our complexity measure to deﬁne a prior distribution over a hypothesis space of theories: P (T ) ∝ 2−c(T ) .4 Given this prior distribution, we can use Bayesian inference to make predictions about unobserved relations and to discover the theory T that best accounts for the observations in data set D [12, 13]. Suppose that we have a likelihood function P (D|T ) which speciﬁes how the examples in D were generated from some underlying theory T . The best explanation for the data D is the theory that maximizes the posterior distribution P (T |D) ∝ P (D|T )P (T ). If we need to predict whether ground term g is likely to be true, 5 we can sum over the space of theories: P (g|D) = P (g|T )P (T |D) = T 1 P (D) P (D|T )P (T ) (2) T :g∈T where the ﬁnal sum is over all theories T that make ground term g true. 1.1 Related work The theories we consider are closely related to logic programs, and methods for Inductive Logic Programming (ILP) explore how these programs can be learned from examples [14]. ILP algorithms are often inspired by the idea of searching for the shortest theory that accounts for the available data, and ILP is occasionally cast as the problem of minimizing an explicit MDL criterion [10]. Although ILP algorithms are rarely considered as cognitive models, the RL approach has a long psychological history, and is proposed by Chomsky [15] and Leeuwenberg [16] among others. Formal measures of complexity have been developed in many ﬁelds [17], and there is at least one other psychological account of theory complexity. Goodman [1] developed a complexity measure that was originally a philosophical proposal about scientiﬁc theories, but was later tested as a model of subjective complexity [6]. A detailed description of this measure is not possible here, but we attempt to give a ﬂavor of the approach. Suppose that a basis is a set of predicates. The starting point for Goodman’s model is the intuition that basis B1 is at least as complex as basis B2 if B1 can be used to deﬁne B2. Goodman argues that this intuition is ﬂawed, but his model is founded on a reﬁnement of this intuition. For instance, since the binary predicate in Fig. 1b can be deﬁned in terms of two unary predicates, Goodman’s approach requires that the complexity of the binary predicate is no more than the sum of the complexities of the two unary predicates. We will use Goodman’s model as a baseline for evaluating our own approach, and a comparison between these two models should be informed by both theoretical and empirical considerations. On the theoretical side, our approach relies on a simple principle for deciding which structural properties are relevant to the measurement of complexity: the relevant properties are those with short logical representations. Goodman’s approach incorporates no such principle, and he proposes somewhat arbitrarily that reﬂexivity and symmetry are among the relevant structural properties but that transitivity is not. A second reason for preferring our model is that it makes contact with a general principle—the idea that simplicity is related to representation length—that has found many applications across psychology, machine learning, and philosophy. 2 Experimental results We designed two experiments to explore settings where people learn, remember, and make inductive inferences about relational data. Although theories often consist of systems of many interlocking relations, we keep our experiments simple by asking subjects to learn and reason about a single relation at a time. Despite this restriction, our experiments still make contact with several issues raised by systems of relations. As the defers to(·, ·) example suggests, a single relation may be best explained as the observable tip of a system involving several latent predicates (e.g. caste 1(·) and caste 2(·)). 4 To ensure that this distribution can be normalized, we assume that there is some upper bound on the number of predicate symbols, variable symbols, and constants, and on the length of the theories we will consider. There will therefore be a ﬁnite number of possible theories, and our prior will be a valid probability distribution. 5 A ground term is a term such as R(8, 9) that does not include any variables. 4 Learning time Complexity (RL) Complexity (Human) 6 300 Complexity (Goodman) 4 20 0 0 0 star bprt excp sym trans rand 2 0 star bprt excp sym trans rand 2 star bprt excp sym trans rand 100 200 star bprt excp sym trans rand 4 40 Figure 2: (a) Average time in seconds to learn the six sets in Fig. 1. (b) Average ratings of set complexity. (c) Complexity scores according to our representation length (RL) model. (d) Complexity scores according to Goodman’s model. 2.1 Experiment 1: memory and induction In our ﬁrst experiment, we studied the subjective complexity of six binary relations that display a range of structural properties, including reﬂexivity, symmetry, and transitivity. Materials and Methods. 18 adults participated in this experiment. Subjects were required to learn the 6 sets shown in Fig. 1, and to make inductive inferences about each set. Although Fig. 1 shows pairs of digits, the experiment used letter pairs, and the letters for each condition and the order in which these conditions were presented were randomized across subjects. The pairs for each condition were initially laid out randomly on screen, and subjects could drag them around and organize them to help them understand the structure of the set. At any stage, subjects could enter a test phase where they were asked to list the 15 pairs belonging to the current set. Subjects who made an error on the test were returned to the learning phase. After 9 minutes had elapsed, subjects were allowed to pass the test regardless of how many errors they made. After passing the test, subjects were asked to rate the complexity of the set compared to other sets with 15 pairs. Ratings were provided on a 7 point scale. Subjects were then asked to imagine that a new letter (e.g. letter 9) had belonged to the current alphabet, and were given two inductive tasks. First they were asked to enter between 1 and 10 novel pairs that they might have expected to see (each novel pair was required to include the new letter). Next they were told about a novel pair that belonged to the set (e.g. pair 91), and were again asked to enter up to 10 additional pairs that they might have expected to see. Results. The average time needed to learn each set is shown in Fig. 2a, and ratings of set complexity are shown in Fig. 2b. It is encouraging that these measures yield converging results, but they may be confounded since subjects rated the complexity of a set immediately after learning it. The complexities plotted in Fig. 2c are the complexities of the theories shown in Fig. 1, which we believe to be the simplest theories according to our complexity measure. The ﬁnal plot in Fig. 2 shows complexities according to Goodman’s model, which assigns each binary relation an integer between 0 and 4. There are several differences between these models: for instance, Goodman’s account incorrectly predicts that the exception case is the hardest of the six, but our model acknowledges that a simple theory remains simple if a handful of exceptions are added. Goodman’s account also predicts that transitivity is not an important structural regularity, but our model correctly predicts that the transitive set is simpler than the same set with some of the pairs reversed (the random set). Results for the inductive task are shown in Fig. 3. The ﬁrst two columns show the number of subjects who listed each novel pair. The remaining two columns show the probability of set membership predicted by our model. To generate these predictions, we applied Equation 2 and summed over a set of theories created by systematically extending the theories shown in Fig. 1. Each extended theory includes up to one additional clause for each predicate in the base theory, and each additional clause includes at most two predicate slots. For instance, each extended theory for the bipartite case is created by choosing whether or not to add the clause T(9), and adding up to one clause for predicate R.6 For the ﬁrst inductive task, the likelihood term P (D|T ) (see Equation 2) is set to 0 for all theories that are not consistent with the pairs observed during training, and to a constant for all remaining theories. For the second task we assumed in addition that the novel pair observed is 6 R(9, X), ¯(2, 9), and R(X, 9) ← R(X, 2) are three possible additions. R 5 18 9 9 0.5 trans symm excep bipart 0 91 random r=0.99 1 star 18 99 19 0 91 89 99 19 89 0.5 0 91 18 18 1 9 9 99 19 r=0.96 89 0.5 0 91 99 19 0 91 89 99 19 89 18 9 1 99 19 89 r=0.98 1 9 0.5 0 91 99 19 0 91 89 99 19 89 18 9 99 19 89 0.5 0 81 88 18 0 78 81 88 18 78 0 18 18 9 0 0 71 77 17 67 71 77 17 67 18 18 81 9 88 18 r=0.62 78 71 77 17 67 Human (no examples) 0 71 77 17 67 Human (1 example) 0 0 91 99 19 89 r=0.99 0 81 88 18 78 71 77 17 67 1 71 77 17 67 r=0.38 0.5 0 89 r=0.93 0.5 1 9 99 19 r=0.99 1 0.5 0 89 r=0.99 0.5 1 9 0 91 1 r=0.88 1 9 99 19 0.5 0 91 18 0 91 0.5 0 91 18 r=0.99 1 0 r=0.74 1 0.5 71 77 17 67 RL (no examples) 0 71 77 17 67 RL (one example) Figure 3: Data and model predictions for the induction task in Experiment 1. Columns 1 and 3 show predictions before any pairs involving the new letter are observed. Columns 2 and 4 show predictions after a single novel pair (marked with a gray bar) is observed to belong to the set. The model plots for each condition include correlations with the human data. sampled at random from all pairs involving the new letter.7 All model predictions were computed using Mace4 [18] to generate the extension of each theory considered. The supporting material includes predictions for a model based on the Goodman complexity measure and an exemplar model which assumes that the new letter will be just like one of the old letters.8 The exemplar model outperforms our model in the random condition, and makes accurate predictions about three other conditions. Overall, however, our model performs better than the two baselines. Here we focus on two important predictions that are not well handled by the exemplar model. In the symmetry condition, almost all subjects predict that 78 belongs to the set after learning that 87 belongs to the set, suggesting that they have learned an abstract rule. In the transitive condition, most subjects predict that pairs 72 through 76 belong to the set after learning that 71 belongs to the set. Our model accounts for this result, but the exemplar model has no basis for making predictions about letter 7, since this letter is now known to be unlike any of the others. 2.2 Experiment 2: learning from positive examples During the learning phase of our ﬁrst experiment, subjects learned a theory based on positive examples (the theory included all pairs they had seen) and negative examples (the theory ruled out all pairs they had not seen). Often, however, humans learn theories based on positive examples alone. Suppose, for instance, that our anthropologist has spent only a few hours with a new tribe. She may have observed several pairs who are obviously friends, but should realize that many other pairs of friends have not yet interacted in her presence. 7 For the second task, P (D|T ) is set to 0 for theories that are inconsistent with the training pairs and theories 1 which do not include the observed novel pair. For all remaining theories, P (D|T ) is set to n , where n is the total number of novel pairs that are consistent with T . 8 Supporting material is available at www.charleskemp.com 6 1 221 331 441 551 c) 7 1 R(X, X, Y). 221 443 552 663 d) 7 1 R(X, Y, Z). 231 456 615 344 e) 7 1 −10 −5 −0.1 −10 −20 −20 −10 −0.2 −20 777 771 778 789 237 777 771 778 789 237 −10 231 234 235 236 0 777 771 778 789 237 0 777 771 778 789 237 0 777 771 778 789 237 0 RL 0 R(2, 3, X). 777 771 778 789 237 7 R(X, X, 1). 777 771 778 789 237 b) 777 771 778 789 237 1 111 222 333 444 777 771 778 789 237 7 R(X, X, X). 777 771 778 789 237 Human a) Figure 4: Data and model predictions for Experiment 2. The four triples observed for each set are shown at the top of the ﬁgure. The ﬁrst row of plots shows average ratings on a scale from 1 (very unlikely to belong to the set) to 7 (very likely). Model predictions are plotted as log probabilities. Our framework can handle cases like these if we assume that the data D in Equation 2 are sampled from the ground terms that are true according to the underlying theory. We follow [10] and [13] and use a distribution P (D|T ) which assumes that the examples in D are randomly sampled with replacement from the ground terms that are true. This sampling assumption encourages our model to identify the theory with the smallest extension that is compatible with all of the training examples. We tested this approach by designing an experiment where learners were given sets of examples that were compatible with several underlying theories. Materials and Methods. 15 adults participated in this experiment immediately after taking Experiment 1. In each of ﬁve conditions, subjects were told about a set of triples built from an alphabet of 9 letters. They were shown four triples that belonged to the set (Fig. 4), and told that the set might include triples that they had not seen. Subjects then gave ratings on a seven point scale to indicate whether ﬁve additional triples (see Fig. 4) were likely to belong to the set. Results. Average ratings and model predictions are shown in Fig. 4. Model predictions for each condition were computed using Equation 2 and summing over a space of theories that included the ﬁve theories shown at the top of Fig. 4, variants of these ﬁve theories which stated that certain pairs of slots could not be occupied by the same constant,9 and theories that included no variables but merely enumerated up to 5 triples.10 Although there are general theories like R(X, Y, Z) that are compatible with the triples observed in all ﬁve conditions, Fig. 4 shows that people were sensitive to different regularities in each case.11 We focus on one condition (Fig. 4b) that exposes the strengths and weaknesses of our model. According to our model, the two most probable theories given the triples for this condition are R(X, X, 1) and the closely related variant that rules out R(1, 1, 1). The next most probable theory is R(X, X, Y). These predictions are consistent with people’s judgments that 771 is very likely to belong to the set, and that 778 is the next most likely option. Unlike our model, however, people consider 777 to be substantially less likely than 778 to belong to the set. This result may suggest that the variant of R(X, X, Y) that rules out R(X, X, X) deserves a higher prior probability than our model recognizes. To better account for cases like this, it may be worth considering languages where any two variables that belong to the same clause but have different names must refer to different entities. 3 Discussion and Conclusion There are many psychological models of concept learning [4, 12, 13], but few that use representations rich enough to capture the content of intuitive theories. We suggested that intuitive theories are mentally represented in a ﬁrst-order logical language, and proposed a speciﬁc hypothesis about 9 ¯ One such theory includes two clauses: R(X, X, Y). R(X, X, X). One such theory is the following list of clauses: R(2, 2, 1). R(3, 3, 1). R(4, 4, 1). R(5, 5, 1). R(7, 7, 7). 11 Similar results have been found with 9-month old infants. Cases like Figs. 4b and 4c have been tested in an infant language-learning study where the stimuli were three-syllable strings [19]. 9-month old infants exposed to strings like the four in Fig. 4c generalized to other strings consistent with the theory R(X, X, Y), but infants in the condition corresponding to Fig. 4b generalized only to strings consistent with the theory R(X, X, 1). 10 7 this “language of thought.” We assumed that the subjective complexity of a theory depends on the length of its representation in this language, and described experiments which suggest that the resulting complexity measure helps to explain how theories are learned and used for inductive inference. Our experiments deliberately used stimuli that minimize the inﬂuence of prior knowledge. Theories, however, are cumulative, and the theory that seems simplest to a learner will often depend on her background knowledge. Our approach provides a natural place for background knowledge to be inserted. A learner can be supplied with a stock of background predicates, and the shortest representation for a data set will depend on which background predicates are available. Since different sets of predicates will lead to different predictions about subjective complexity, empirical results can help to determine the background knowledge that people bring to a given class of problems. Future work should aim to reﬁne the representation language and complexity measure we proposed. We expect that something like our approach will be suitable for modeling a broad class of intuitive theories, but the speciﬁc framework presented here can almost certainly be improved. Future work should also consider different strategies for searching the space of theories. Some of the strategies developed in the ILP literature should be relevant [14], but a detailed investigation of search algorithms seems premature until our approach has held up to additional empirical tests. It is comparatively easy to establish whether the theories that are simple according to our approach are also considered simple by people, and our experiments have made a start in this direction. It is much harder to establish that our approach captures most of the theories that are subjectively simple, and more exhaustive experiments are needed before this conclusion can be drawn. Boolean concept learning has been studied for more than ﬁfty years [4, 9], and many psychologists have made empirical and theoretical contributions to this ﬁeld. An even greater effort will be needed to crack the problem of theory learning, since the space of intuitive theories is much richer than the space of Boolean concepts. The difﬁculty of this problem should not be underestimated, but computational approaches can contribute part of the solution. Acknowledgments Supported by the William Asbjornsen Albert memorial fellowship (CK), the James S. McDonnell Foundation Causal Learning Collaborative Initiative (NDG, JBT) and the Paul E. Newton chair (JBT). References [1] N. Goodman. The structure of appearance. 2nd edition, 1961. [2] S. Carey. Conceptual change in childhood. MIT Press, Cambridge, MA, 1985. [3] H. A. Simon. Complexity and the representation of patterned sequences of symbols. Psychological Review, 79:369–382, 1972. [4] J. Feldman. An algebra of human concept learning. JMP, 50:339–368, 2006. [5] N. Chater and P. Vitanyi. Simplicity: a unifying principle in cognitive science. TICS, 7:19–22, 2003. [6] J. T. Krueger. A theory of structural simplicity and its relevance to aspects of memory, perception, and conceptual naturalness. PhD thesis, University of Pennsylvania, 1979. [7] P. Gr¨ nwald, I. J. Myung, and M. Pitt, editors. Advances in Minimum Description Length: Theory and u Applications. 2005. [8] N. Chater. Reconciling simplicity and likelihood principles in perceptual organization. Psychological Review, 103:566–581, 1996. [9] J. A. Bruner, J. S. Goodnow, and G. J. Austin. A study of thinking. Wiley, 1956. [10] D. Conklin and I. H. Witten. Complexity-based induction. Machine Learning, 16(3):203–225, 1994. [11] G. A. Miller. The magical number seven, plus or minus two: Some limits on our capacity for processing information. Psychological Review, 63(1):81–97, 1956. [12] N. D. Goodman, T. L. Grifﬁths, J. Feldman, and J. B. Tenenbaum. A rational analysis of rule-based concept learning. In CogSci, 2007. [13] J. B. Tenenbaum and T. L. Grifﬁths. Generalization, similarity, and Bayesian inference. BBS, 24:629–641, 2001. [14] S. Muggleton and L. De Raedt. Inductive logic programming: theory and methods. Journal of Logic Programming, 19-20:629–679, 1994. [15] N. Chomsky. The logical structure of linguistic theory. University of Chicago Press, Chicago, 1975. [16] E. L. J. Leeuwenberg. A perceptual coding language for visual and auditory patterns. American Journal of Psychology, 84(3):307–349, 1971. [17] B. Edmonds. Syntactic measures of complexity. PhD thesis, University of Manchester, 1999. [18] W. McCune. Mace4 reference manual and guide. Technical Report ANL/MCS-TM-264, Argonne National Laboratory, 2003. [19] L. Gerken. Decisions, decisions: infant language learning when multiple generalizations are possible. Cognition, 98(3):67–74, 2006. 8

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