Structure detection: a statistically certified unsupervised learning procedure

We present a class of structure detection procedures (SDPs) that can extract the characteristic structures in an arbitrary population of images. An SDP adaptively augments the power of a novel, statistical, structure test to reject the null hypothesis that a randomly chosen image is devoid of structure. The core of the structure test consists of an orthonormal basis B of receptive fields that is refined into an increasingly sensitive detector of characteristic image structures. Adaptive refinement is accomplished as follows: for each image x in a random training sequence, B is updated by a planar rotation that decreases the p-value of a statistical structure test for x. This image-by-image refinement procedure is very efficient, obeying time and space constraints similar to those that limit processes of perceptual organization in real organisms. SDPs' capabilities are demonstrated in three test populations: natural images, faulty random number generators, and artificial images composed of mixtures of basis functions. (1) An SDP succeeds in rejecting the null hypothesis that the UNIX random number generator rand() is truly random. (2) When images are composed by adding arbitrary pairs of orthogonal component images, an SDP extracts the components. (3) For a large set of natural image patches, an SDP yields a basis B1 that detects structure with p-value < 0.005 in 88% of a new set of patches. B1's elements resemble the receptive fields of V1 simple cells. (4) Of special interest are biconvergent SDPs that derive in parallel a basis B, as well as a pointwise transformation f, specifically sensitized to evaluate the response values that result from applying B to images in the target population. A biconvergent SDP applied to natural image patches yields a basis B2 similar to B1, as well as a pointwise transformation f with vastly heightened sensitivity to extreme response values. We conjecture that sensory neurons have evolved cooperatively to maximize their collective power to reject the null hypothesis that their input is devoid of structure, thereby evolving receptive fields that efficiently represent characteristic input structures.