Learning Invariant Riemannian Geometric Representations Using Deep Nets

Non-Euclidean constraints are inherent in many kinds of data in computer vision and machine learning, typically as a result of specific invariance requirements that need to be respected during high-level inference. Often, these geometric constraints can be expressed in the language of Riemannian geometry, where conventional vector space machine learning does not apply directly. The central question this paper deals with is: How does one train deep neural nets whose final outputs are elements on a Riemannian manifold? To answer this, we propose a general framework for manifold-aware training of deep neural networks -- we utilize tangent spaces and exponential maps in order to convert the proposed problem into a form that allows us to bring current advances in deep learning to bear upon this problem. We describe two specific applications to demonstrate this approach: prediction of probability distributions for multi-class image classification, and prediction of illumination-invariant subspaces from a single face-image via regression on the Grassmannian. These applications show the generality of the proposed framework, and result in improved performance over baselines that ignore the geometry of the output space. In addition to solving this specific problem, we believe this paper opens new lines of enquiry centered on the implications of Riemannian geometry on deep architectures.Comment: Accepted at International Conference on Computer Vision Workshop (ICCVW), 2017 on Manifold Learning: from Euclid to Rieman

DISCOV: A Neural Model of Colour Vision, with Applications to Image Processing and Classification

The DISCOV (Dimensionless Shunting Colour Vision) system models a cascade of primate colour vision cells: retinal ganglion, thalamic single opponent, and two classes of cortical double opponents. A unified model fotmalism derived from psychophysical axioms produces transparent network dynamics and principled parameter settings. DISCOV fits an array of physiological data for each cell type, and makes testable experimental predictions. Properties of DISCOV model cells are compared with properties of conesponding components in the alternative Neural Fusion model. A benchmark testbed demonstrates the marginal computational utility of each model cell type on a recognition task derived from orthophoto imagery.Air Force Office of Scientific Research (F49620-01-1-0423); National Geospatial-Intelligence Agency (NMA 201-01-1-2016); National Science Foundation (SBE-035437, DGE-0221680); Office of Naval Research (N00014-01-1-0624

CEDI: A Neural Model of Colour Vision, with Applications to Image Processing and Classification

Air Force Office of Scientific Research (F49620-01-1-0423); National Geospatial-Intelligence Agency (NMA 201-01-1-2016); National Science Foundation (SBE-035437, DEG-0221680); Office of Naval Research (N00014-01-1-0624

The Approximate Capacity of the Gaussian N-Relay Diamond Network

We consider the Gaussian "diamond" or parallel relay network, in which a source node transmits a message to a destination node with the help of N relays. Even for the symmetric setting, in which the channel gains to the relays are identical and the channel gains from the relays are identical, the capacity of this channel is unknown in general. The best known capacity approximation is up to an additive gap of order N bits and up to a multiplicative gap of order N^2, with both gaps independent of the channel gains. In this paper, we approximate the capacity of the symmetric Gaussian N-relay diamond network up to an additive gap of 1.8 bits and up to a multiplicative gap of a factor 14. Both gaps are independent of the channel gains and, unlike the best previously known result, are also independent of the number of relays N in the network. Achievability is based on bursty amplify-and-forward, showing that this simple scheme is uniformly approximately optimal, both in the low-rate as well as in the high-rate regimes. The upper bound on capacity is based on a careful evaluation of the cut-set bound. We also present approximation results for the asymmetric Gaussian N-relay diamond network. In particular, we show that bursty amplify-and-forward combined with optimal relay selection achieves a rate within a factor O(log^4(N)) of capacity with pre-constant in the order notation independent of the channel gains.Comment: 23 pages, to appear in IEEE Transactions on Information Theor

On Multistage Successive Refinement for Wyner-Ziv Source Coding with Degraded Side Informations

We provide a complete characterization of the rate-distortion region for the multistage successive refinement of the Wyner-Ziv source coding problem with degraded side informations at the decoder. Necessary and sufficient conditions for a source to be successively refinable along a distortion vector are subsequently derived. A source-channel separation theorem is provided when the descriptions are sent over independent channels for the multistage case. Furthermore, we introduce the notion of generalized successive refinability with multiple degraded side informations. This notion captures whether progressive encoding to satisfy multiple distortion constraints for different side informations is as good as encoding without progressive requirement. Necessary and sufficient conditions for generalized successive refinability are given. It is shown that the following two sources are generalized successively refinable: (1) the Gaussian source with degraded Gaussian side informations, (2) the doubly symmetric binary source when the worse side information is a constant. Thus for both cases, the failure of being successively refinable is only due to the inherent uncertainty on which side information will occur at the decoder, but not the progressive encoding requirement.Comment: Submitted to IEEE Trans. Information Theory Apr. 200