Invoking the manifold assumption in machine learning requires knowledge of the manifold's geometry and dimension, and theory dictates how many samples are required. However, in most applications, the data are limited, sampling may not be uniform, and the manifold's properties are unknown; this implies that neighborhoods must adapt to the local structure. We introduce an algorithm for inferring adaptive neighborhoods for data given by a similarity kernel. Starting with a locally conservative neighborhood (Gabriel) graph, we sparsify it iteratively according to a weighted counterpart. In each step, a linear program yields minimal neighborhoods globally, and a volumetric statistic reveals neighbor outliers likely to violate manifold geometry. We apply our adaptive neighborhoods to nonlinear dimensionality reduction, geodesic computation, and dimension estimation. A comparison against standard algorithms using, for example, k -nearest neighbors, demonstrates the usefulness of our approach.

We present a model of visual computation based on tightly inter-connected cliques of pyramidal cells. It leads to a formal theory of cell assemblies, a specific relationship between correlated firing patterns and abstract functionality, and a direct calculation relating estimates of cortical cell counts to orientation hyperacuity. Our network architecture is unique in that (1) it supports a mode of computation that is both reliable and efficent; (2) the current-spike relations are modeled as an analog dynamical system in which the requisite computations can take place on the time scale required for an early stage of visual processing; and (3) the dynamics are triggered by the spatiotemporal response of cortical cells. This final point could explain why moving stimuli improve vernier sensitivity.

What is the complexity of computing equilibria for physically implementable analog networks (Hopfield 1984; Sejnowski 1981) with arbitrary connectivity? We show that if the amplifiers are piecewise-linear, then such networks are instances of a game-theoretic model known as polymatrix games . In contrast with the usual gradient descent methods for symmetric networks, equilibria for polymatrix games may be computed by vertex pivoting algorithms similar to the simplex method for linear programming. Like the simplex method, these algorithms have characteristic low order polynomial behavior in virtually all practical cases, though not certain theoretical ones. While these algorithms cannot be applied to models requiring evolution from an initial point, they are applicable to “clamping” models whose input is expressed purely as a bias. Thus we have an a priori indication that such models are computationally tractable.

Consider two wire gratings, superimposed and moving across each other. Under certain conditions the two gratings will cohere into a single, compound pattern, which will appear to be moving in another direction. Such coherent motion patterns have been studied for sinusoidal component gratings, and give rise to percepts of rigid, planar motions. In this paper we show how to construct coherent motion displays that give rise to nonuniform, nonrigid, and nonplanar percepts. Most significantly, they also can define percepts with corners. Since these patterns are more consistent with the structure of natural scenes than rigid sinusoidal gratings, they stand as interesting stimuli for both computational and physiological studies. To illustrate, our display with sharp corners (tangent discontinuities or singularities) separating regions of coherent motion suggests that smoothing does not cross tangent discontinuities, a point that argues against existing (regularization) algorithms for computing motion. This leads us to consider how singularities can be confronted directly within optical flow computations, and we conclude with two hypotheses: (1) that singularities are represented within the motion system as multiple directions at the same retinotopic location; and (2) for component gratings to cohere, they must be at the same depth from the viewer. Both hypotheses have implications for the neural computation of coherent motion.

The problem of detecting curves in visual images arises in both computer vision and biological visual systems. Our approach integrates constraints from these two sources and suggests that there are two different stages to curve detection, the first resulting in a local description, and the second in a global one. Each stage involves a different style of computation: in the first stage, hypotheses are represented explicitly and coarsely in a fixed, preconfigured architecture; in the second stage, hypotheses are represented implicitly and more finely in a dynamically constructed architecture. We also show how these stages could be related to physiology, specifying the earlier parts in a relatively fine-grained fashion and the later ones more coarsely.