Jacob Feldman

Minimization of Boolean complexity in human concept learning

One of the unsolved problems in the field of human concept learning concerns the factors that determine the subjective difficulty of concepts: why are some concepts psychologically simple and easy to learn, while others seem difficult, complex or incoherent? This question was much studied in the 1960s but was never answered, and more recent characterizations of concepts as prototypes rather than logical rules leave it unsolved. Here I investigate this question in the domain of Boolean concepts (categories defined by logical rules). A series of experiments measured the subjective difficulty of a wide range of logical varieties of concepts (41 mathematically distinct types in six families—a far wider range than has been tested previously). The data reveal a surprisingly simple empirical ‘law’: the subjective difficulty of a concept is directly proportional to its Boolean complexity (the length of the shortest logically equivalent propositional formula)—that is, to its logical incompressibility.

A Rational Analysis of Rule-based Concept Learning

This article proposes a new model of human concept learning that provides a rational analysis of learning feature-based concepts. This model is built upon Bayesian inference for a grammatically structured hypothesis space-a concept language of logical rules. This article compares the model predictions to human generalization judgments in several well-known category learning experiments, and finds good agreement for both average and individual participant generalizations. This article further investigates judgments for a broad set of 7-feature concepts-a more natural setting in several ways-and again finds that the model explains human performance.

Information along contours and object boundaries.

F. Attneave (1954) famously suggested that information along visual contours is concentrated in regions of high magnitude of curvature, rather than being distributed uniformly along the contour. Here the authors give a formal derivation of this claim, yielding an exact expression for information, in C. Shannons (1948) sense, as a function of contour curvature. Moreover, they extend Attneaves claim to incorporate the role of sign of curvature, not just magnitude of curvature. In particular, the authors show that for closed contours, such as object boundaries, segments of negative curvature (i.e., concave segments) literally carry greater information than do corresponding regions of positive curvature (i.e., convex segments). The psychological validity of this informational analysis is supported by a host of empirical findings demonstrating the asymmetric way in which the visual system treats regions of positive and negative curvature.

A Century of Gestalt Psychology in Visual Perception: II. Conceptual and Theoretical Foundations.

Our first review article (Wagemans et al., 2012) on the occasion of the centennial anniversary of Gestalt psychology focused on perceptual grouping and figure-ground organization. It concluded that further progress requires a reconsideration of the conceptual and theoretical foundations of the Gestalt approach, which is provided here. In particular, we review contemporary formulations of holism within an information-processing framework, allowing for operational definitions (e.g., integral dimensions, emergent features, configural superiority, global precedence, primacy of holistic/configural properties) and a refined understanding of its psychological implications (e.g., at the level of attention, perception, and decision). We also review 4 lines of theoretical progress regarding the law of Prägnanz-the brains tendency of being attracted towards states corresponding to the simplest possible organization, given the available stimulation. The first considers the brain as a complex adaptive system and explains how self-organization solves the conundrum of trading between robustness and flexibility of perceptual states. The second specifies the economy principle in terms of optimization of neural resources, showing that elementary sensors working independently to minimize uncertainty can respond optimally at the system level. The third considers how Gestalt percepts (e.g., groups, objects) are optimal given the available stimulation, with optimality specified in Bayesian terms. Fourth, structural information theory explains how a Gestaltist visual system that focuses on internal coding efficiency yields external veridicality as a side effect. To answer the fundamental question of why things look as they do, a further synthesis of these complementary perspectives is required.

Bayesian estimation of the shape skeleton

Skeletal representations of shape have attracted enormous interest ever since their introduction by Blum [Blum H (1973) J Theor Biol 38:205–287], because of their potential to provide a compact, but meaningful, shape representation, suitable for both neural modeling and computational applications. But effective computation of the shape skeleton remains a notorious unsolved problem; existing approaches are extremely sensitive to noise and give counterintuitive results with simple shapes. In conventional approaches, the skeleton is defined by a geometric construction and computed by a deterministic procedure. We introduce a Bayesian probabilistic approach, in which a shape is assumed to have “grown” from a skeleton by a stochastic generative process. Bayesian estimation is used to identify the skeleton most likely to have produced the shape, i.e., that best “explains” it, called the maximum a posteriori skeleton. Even with natural shapes with substantial contour noise, this approach provides a robust skeletal representation whose branches correspond to the natural parts of the shape.

The influence of spatial context and the role of intentionality in the interpretation of animacy from motion

We present three experiments investigating how spatial context influences the attribution of animacy to a moving target. Each of our displays contained a moving object (the target) that might, depending on the way it moved, convey the impression that it was alive (animate). We investigated the mechanisms underlying this attribution by manipulating the nature of the spatial context surrounding the target. In Experiment 1, the context consisted of a simple static dot (the foil), whose position relative to the target’s trajectory was manipulated. With some foil positions—for example, when the foil was lying along the path traveled by the target—animacy judgments were elevated relative to control foil locations, apparently because this context supported the impression that the target was “reacting to” or was in some other way mentally influenced by the foil. In Experiment 2, contexts consisted of a static oriented rectangle (the “paddle”). On some trials, the target collided with the paddle in a way that seemed to physically account for the target’s motion pattern (in the sense of having imparted momentum to it); this condition reduced animacy ratings. Experiment 3 was similar, except that the paddles themselves were in motion; again, animacy attribution was suppressed when the target’s motion seemed to have been caused by a collision with the paddle. Hence, animacy attributions can be either elevated or suppressed by the nature of the environment and the target’s interaction with it. Animacy attribution tracks intentionality attribution; contrary to some earlier proposals, we conclude that attributing animacy involves, and may even require, attributing to the target some minimal mental capacity sufficient to endow the target with intentionality.

Bayesian contour integration

The process by which the human visual system parses an image into contours, surfaces, and objects—perceptual grouping—has proven difficult to capture in a rigorous and general theory. A natural candidate for such a theory is Bayesian probability theory, which provides optimal interpretations of data under conditions of uncertainty. But the fit of Bayesian theory to human grouping judgments has never been tested, in part because methods for expressing grouping hypotheses probabilistically have not been available. This paper presents such methods for the case ofcontour integration—that is, the aggregation of a sequence of visual items into a “virtual curve.” Two experiments are reported in which human subjects were asked to group ambiguous configurations of dots (in Experiment 1, a sequence of five dots could be judged to contain a “corner” or not; in Experiment 2, an arrangement of six dots could be judged to fall into two disjoint contours or one smooth contour). The Bayesian theory accounts extremely well for subjects’ judgments, explaining more than 75% of the variance in both tasks. The theory thus provides a far more quantitatively precise account of human contour integration than has been previously possible, allowing a very precise calculation of the subjective goodness of a virtual chain of dots. Because Bayesian theory is inferentially optimal, this finding suggests a “rational justification,” and hence possibly an evolutionary rationale, for some of the rules of perceptual grouping.

Curvilinearity, covariance, and regularity in perceptual groups

A curvilinear pattern among a series of visual items (e.g., dots) can be regarded as a kind of probabilistic inference, in which each consecutive angle, regarded independently, is more nearly collinear than would be expected by chance alone. This paper investigates judgments of curvilinearity as a function of the joint distribution of successive inter-dot angles. Subjects were asked to classify 4- and 5-dot configurations as having been generated by a curvilinear generating process, vs independently. Their results are distributed as a gaussian over the inter-dot angles with mean 0 deg (collinear), with a negative correlation between successive angles, but negligible correlation between non-successive angles. This suggests that curvilinearity is evaluated in a 4-dot window moving along the chain of dots, evaluating collinearity and smoothness but ignoring higher-order relationships. Moreover, the probabilistic model provides a remarkably precise numeric prediction of the magnitude of the correlation. Subjects also showed a reliable preference for equal spacing of dots along the virtual curve.

Detection of change in shape: an advantage for concavities

Shape representation was studied using a change detection task. Observers viewed two individual shapes in succession, either identical or one a slightly altered version of the other, and reported whether they detected a change. We found a dramatic advantage for concave compared to convex changes of equal magnitude. Observers were more accurate when a concavity along the contour was introduced, or removed, compared to a convexity. This result sheds light on the underlying representation of visual shape, and in particular the central role played by part-boundaries. Moreover, this finding shows how change detection methodology can serve as a useful tool in studying the specific form of visual representations.

Formation of visual "objects" in the early computation of spatial relations.

Perceptual grouping is the process by which elements in the visual image are aggregated into larger and more complex structures, i.e., “objects.” This paper reports a study of the spatial factors and time-course of the development of objects over the course of the first few hundred milliseconds of visual processing. The methodology uses the now well-established idea of an “object benefit” for certain kinds of tasks (here, faster within-object than between-objects probe comparisons) to test what the visual system in fact treats as an object at each point during processing. The study tested line segment pairs in a wide variety of spatial configurations at a range of exposure times, in each case measuring the strength of perceptual grouping as reflected in the magnitude of the object benefit. Factors tested included nonaccidental properties such as collinearity, cotermination, and parallelism; contour relatability; Gestalt factors such as symmetry and skew symmetry, and several others, all tested at fine (25 msec) time-slices over the course of processing. The data provide detailed information about the comparative strength of these factors in inducing grouping at each point in processing. The result is a vivid picture of the chronology of object formation, as objects progressively coalesce, with fully bound visual objects completed by about 200 msec of processing.