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We model the functional architecture of the early stages of three-dimensional vision by extending the neurogeometric sub-Riemannian model for stereo-vision introduced in \cite{BCSZ23}. A new framework for correspondence is introduced that integrates a neural-based algorithm to achieve stereo correspondence locally while, simultaneously, organizing the corresponding points into global perceptual units. The result is an effective scene segmentation. We achieve this using harmonic analysis on the sub-Riemannian structure and show, in a comparison against Riemannian distance, that the sub-Riemannian metric is central to the solution., Comment: 30 pages, 13 figures

Generalization to unseen data remains poorly understood for deep learning classification and foundation models, especially in the open set scenario. How can one assess the ability of networks to adapt to new or extended versions of their input space in the spirit of few-shot learning, out-of-distribution generalization, domain adaptation, and category discovery? Which layers of a network are likely to generalize best? We provide a new method for evaluating the capacity of networks to represent a sampled domain, regardless of whether the network has been trained on all classes in that domain. Our approach is the following: after fine-tuning state-of-the-art pre-trained models for visual classification on a particular domain, we assess their performance on data from related but distinct variations in that domain. Generalization power is quantified as a function of the latent embeddings of unseen data from intermediate layers for both unsupervised and supervised settings. Working throughout all stages of the network, we find that (i) high classification accuracy does not imply high generalizability; and (ii) deeper layers in a model do not always generalize the best, which has implications for pruning. Since the trends observed across datasets are largely consistent, we conclude that our approach reveals (a function of) the intrinsic capacity of the different layers of a model to generalize. Our code is available at https://github.com/dyballa/generalization, Comment: 7 pages, 6 figures

The static synaptic connectivity of neuronal circuits stands in direct contrast to the dynamics of their function. As in changing community interactions, different neurons can participate actively in various combinations to effect behaviors at different times. We introduce an unsupervised approach to learn the dynamic affinities between neurons in live, behaving animals, and to reveal which communities form among neurons at different times. The inference occurs in two major steps. First, pairwise non-linear affinities between neuronal traces from brain-wide calcium activity are organized by non-negative tensor factorization (NTF). Each factor specifies which groups of neurons are most likely interacting for an inferred interval in time, and for which animals. Finally, a generative model that allows for weighted community detection is applied to the functional motifs produced by NTF to reveal a dynamic functional connectome. Since time codes the different experimental variables (e.g., application of chemical stimuli), this provides an atlas of neural motifs active during separate stages of an experiment (e.g., stimulus application or spontaneous behaviors). Results from our analysis are experimentally validated, confirming that our method is able to robustly predict causal interactions between neurons to generate behavior. Code is available at https://github.com/dyballa/dynamic-connectomes., Comment: Accepted at ICLR 24

Generalization to unseen data is a key desideratum for deep networks, but its relation to classification accuracy is unclear. Using a minimalist vision dataset and a measure of generalizability, we show that popular networks, from deep convolutional networks (CNNs) to transformers, vary in their power to extrapolate to unseen classes both across layers and across architectures. Accuracy is not a good predictor of generalizability, and generalization varies non-monotonically with layer depth., Comment: Accepted as a Tiny Paper at ICLR 2024. Code available at https://github.com/dyballa/generalization/tree/ICLR2024TinyPaper

Classical good continuation for image curves is based on $2D$ position and orientation. It is supported by the columnar organization of cortex, by psychophysical experiments, and by rich models of (differential) geometry. Here we extend good continuation to stereo. We introduce a neurogeometric model, in which the parametrizations involve both spatial and orientation disparities. Our model provides insight into the neurobiology, suggesting an implicit organization for neural interactions and a well-defined $3D$ association field. Our model sheds light on the computations underlying the correspondence problem, and illustrates how good continuation in the world generalizes good continuation in the plane.

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 applications data are limited, sampling may not be uniform, and manifold properties are unknown and (possibly) non-pure; 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 non-linear dimensionality reduction, geodesic computation and dimension estimation. A comparison against standard algorithms using, e.g., k-nearest neighbors, demonstrates their usefulness. Code for our algorithm will be available at https://github.com/dyballa/IAN, Comment: 69 pages, 35 figures

Shape inference is classically ill-posed, because it involves a map from the (2D) image domain to the (3D) world. Standard approaches regularize this problem by either assuming a prior on lighting and rendering or restricting the domain, and develop differential equations or optimization solutions. While elegant, the solutions that emerge in these situations are remarkably fragile. We exploit the observation that people infer shape qualitatively; that there are quantitative differences between individuals. The consequence is a topological approach based on critical contours and the Morse-Smale complex. This paper provides a developmental review of that theory, emphasizing the motivation at different stages of the research.

Invariants underlying shape inference are elusive: a variety of shapes can give rise to the same image, and a variety of images can be rendered from the same shape. The occluding contour is a rare exception: it has both image salience, in terms of isophotes, and surface meaning, in terms of surface normal. We relax the notion of occluding contour to define closed extremal curves, a new shape invariant that exists at the topological level. They surround bumps, a common but ill-specified interior shape component, and formalize the qualitative nature of bump perception. Extremal curves are biologically computable, unify shape inferences from shading, texture, and specular materials, and predict new phenomena in bump perception.

Feature selection can facilitate the learning of mixtures of discrete random variables as they arise, e.g. in crowdsourcing tasks. Intuitively, not all workers are equally reliable but, if the less reliable ones could be eliminated, then learning should be more robust. By analogy with Gaussian mixture models, we seek a low-order statistical approach, and here introduce an algorithm based on the (pairwise) mutual information. This induces an order over workers that is well structured for the `one coin' model. More generally, it is justified by a goodness-of-fit measure and is validated empirically. Improvement in real data sets can be substantial.

We exploit a key result from visual psychophysics---that individuals perceive shape qualitatively---to develop the use of a geometrical/topological "invariant'' (the Morse--Smale complex) relating image structure with surface structure. Differences across individuals are minimal near certain configurations such as ridges and boundaries, and it is these configurations that are often represented in line drawings. In particular, we introduce a method for inferring a qualitative three-dimensional shape from shading patterns that link the shape-from-shading inference with shape-from-contour inference. For a given shape, certain shading patches approach "line drawings'' in a well-defined limit. Under this limit, and invariably with respect to rendering choices, these shading patterns provide a qualitative description of the surface. We further show that, under this model, the contours partition the surface into meaningful parts using the Morse--Smale complex. These critical contours are the (perceptually) stable parts of this complex and are invariant over a wide class of rendering models. Intuitively, our main result shows that critical contours partition smooth surfaces into bumps and valleys, in effect providing a scaffold on the image from which a full surface can be interpolated., Comment: Version 2: A reorganized version with the same results. Technical details are placed in the Appendix

We develop a linear algebraic framework for the shape-from-shading problem, because tensors arise when scalar (e.g. image) and vector (e.g. surface normal) fields are differentiated multiple times. The work is in two parts. In this first part we investigate when image derivatives exhibit invariance to changing illumination by calculating the statistics of image derivatives under general distributions on the light source. We computationally validate the hypothesis that image orientations (derivatives) provide increased invariance to illumination by showing (for a Lambertian model) that a shape-from-shading algorithm matching gradients instead of intensities provides more accurate reconstructions when illumination is incorrectly estimated under a flatness prior.

We develop a linear algebraic framework for the shape-from-shading problem, because tensors arise when scalar (e.g. image) and vector (e.g. surface normal) fields are differentiated multiple times. Using this framework, we first investigate when image derivatives exhibit invariance to changing illumination by calculating the statistics of image derivatives under general distributions on the light source. Second, we apply that framework to develop Taylor-like expansions, and build a boot-strapping algorithm to find the polynomial surface solutions (under any light source) consistent with a given patch to arbitrary order. A generic constraint on the light source restricts these solutions to a 2-D subspace, plus an unknown rotation matrix. It is this unknown matrix that encapsulates the ambiguity in the problem. Finally, we use the framework to computationally validate the hypothesis that image orientations (derivatives) provide increased invariance to illumination by showing (for a Lambertian model) that a shape-from-shading algorithm matching gradients instead of intensities provides more accurate reconstructions when illumination is incorrectly estimated under a flatness prior., Comment: arXiv admin note: substantial text overlap with arXiv:1705.05885

Many organisms navigate gradients by alternating straight motions (runs) with random reorientations (tumbles), transiently suppressing tumbles whenever attractant signal increases. This induces a functional coupling between movement and sensation, since tumbling probability is controlled by the internal state of the organism which, in turn, depends on previous signal levels. Although a negative feedback tends to maintain this internal state close to adapted levels, positive feedback can arise when motion up the gradient reduces tumbling probability, further boosting drift up the gradient. Importantly, such positive feedback can drive large fluctuations in the internal state, complicating analytical approaches. Previous studies focused on what happens when the negative feedback dominates the dynamics. By contrast, we show here that there is a large portion of physiologically-relevant parameter space where the positive feedback can dominate, even when gradients are relatively shallow. We demonstrate how large transients emerge because of non-normal dynamics (non-orthogonal eigenvectors near a stable fixed point) inherent in the positive feedback, and further identify a fundamental nonlinearity that strongly amplifies their effect. Most importantly, this amplification is asymmetric, elongating runs in favorable directions and abbreviating others. The result is a "ratchet-like" gradient climbing behavior with drift speeds that can approach half the maximum run speed of the organism. Our results thus show that the classical drawback of run-and-tumble navigation --- wasteful runs in the wrong direction --- can be mitigated by exploiting the non-normal dynamics implicit in the run-and-tumble strategy., Comment: 22 pages, 3 figures, and 8 pages of SI

The performance of EM in learning mixtures of product distributions often depends on the initialization. This can be problematic in crowdsourcing and other applications, e.g. when a small number of 'experts' are diluted by a large number of noisy, unreliable participants. We develop a new EM algorithm that is driven by these experts. In a manner that differs from other approaches, we start from a single mixture class. The algorithm then develops the set of 'experts' in a stagewise fashion based on a mutual information criterion. At each stage EM operates on this subset of the players, effectively regularizing the E rather than the M step. Experiments show that stagewise EM outperforms other initialization techniques for crowdsourcing and neurosciences applications, and can guide a full EM to results comparable to those obtained knowing the exact distribution., Comment: 9 pages

Shape from shading is a classical inverse problem in computer vision. This shape reconstruction problem is inherently ill-defined; it depends on the assumed light source direction. We introduce a novel mathematical formulation for calculating local surface shape based on covariant derivatives of the shading flow field, rather than the customary integral minimization or P.D.E approaches. On smooth surfaces, we show second derivatives of brightness are independent of the light sources and can be directly related to surface properties. We use these measurements to define the matching local family of surfaces that can result from any given shading patch, changing the emphasis to characterizing ambiguity in the problem. We give an example of how these local surface ambiguities collapse along certain image contours and how this can be used for the reconstruction problem., Comment: 34 pages, 12 figures

The principles underlying plant development are extended to allow a more molecular mechanism to elaborate the schema by which ground cells differentiate into vascular cells. Biophysical considerations dictate that linear dynamics are not sufficent to capture facilitated auxin transport (e.g., through PIN). We group these transport facilitators into a non-linear model under the assumption that they attempt to minimize certain {\em differences} of auxin concentration. This Constant Gradient Hypothesis greatly increases the descriptive power of our model to include complex dynamical behaviour. Specifically, we show how the early pattern of PIN1 expression appears in the embryo, how the leaf primordium emerges, how convergence points arise on the leaf margin, how the first loop is formed, and how the intricate pattern of PIN shifts during the early establishment of vein patterns in incipient leaves of Arabidopsis. Given our results, we submit that the model provides evidence that many of the salient structural characteristics that have been described at various stages of plant development can arise from the uniform application of a small number of abstract principles.

Experimental data regarding auxin and venation formation exist at both macroscopic and molecular scales, and we attempt to unify them into a comprehensive model for venation formation. We begin with a set of principles to guide an abstract model of venation formation, from which we show how patterns in plant development are related to the representation of global distance information locally as cellular-level signals. Venation formation, in particular, is a function of distances between cells and their locations. The first principle, that auxin is produced at a constant rate in all cells, leads to a (Poisson) reaction-diffusion equation. Equilibrium solutions uniquely codify information about distances, thereby providing cells with the signal to begin differentiation from ground to vascular. A uniform destruction hypothesis and scaling by cell size leads to a more biologically-relevant (Helmholtz) model, and simulations demonstrate its capability to predict leaf and root auxin distributions and venation patterns. The mathematical development is centered on properties of the distance map, and provides a mechanism by which global information about shape can be presented locally to individual cells. The principles provide the foundation for an elaboration of these models in a companion paper \cite{plos-paper2}, and together they provide a framework for understanding organ- and plant-scale organization., Comment: pdfLatex