Rodents navigating in a well-known environment can rapidly learn and revisit observed reward locations, often after a single trial. While the mechanism for rapid path planning is unknown, the CA3 region in the hippocampus plays an important role, and emerging evidence suggests that place cell activity during hippocampal preplay periods may trace out future goal-directed trajectories. Here, we show how a particular mapping of space allows for the immediate generation of trajectories between arbitrary start and goal locations in an environment, based only on the mapped representation of the goal. We show that this representation can be implemented in a neural attractor network model, resulting in bump--like activity profiles resembling those of the CA3 region of hippocampus. Neurons tend to locally excite neurons with similar place field centers, while inhibiting other neurons with distant place field centers, such that stable bumps of activity can form at arbitrary locations in the environment. The network is initialized to represent a point in the environment, then weakly stimulated with an input corresponding to an arbitrary goal location. We show that the resulting activity can be interpreted as a gradient ascent on the value function induced by a reward at the goal location. Indeed, in networks with large place fields, we show that the network properties cause the bump to move smoothly from its initial location to the goal, around obstacles or walls. Our results illustrate that an attractor network with hippocampal-like attributes may be important for rapid path planning.

We propose the infinite factorial dynamic model (iFDM), a general Bayesian nonparametric model for source separation. Our model builds on the Markov Indian buffet process to consider a potentially unbounded number of hidden Markov chains (sources) that evolve independently according to some dynamics, in which the state space can be either discrete or continuous. For posterior inference, we develop an algorithm based on particle Gibbs with ancestor sampling that can be efficiently applied to a wide range of source separation problems. We evaluate the performance of our iFDM on four well-known applications: multitarget tracking, cocktail party, power disaggregation, and multiuser detection. Our experimental results show that our approach for source separation does not only outperform previous approaches, but it can also handle problems that were computationally intractable for existing approaches.

We study Nyström type subsampling approaches to large scale kernel methods, and prove learning bounds in the statistical learning setting, where random sampling andhigh probability estimates are considered. In particular, we prove that these approaches can achieve optimal learning bounds, provided the subsampling level is suitably chosen. These results suggest a simple incremental variant of Nyström Kernel Regularized Least Squares, where the subsampling level implements aform of computational regularization, in the sense that it controls at the same time regularization and computations. Extensive experimental analysis showsthat the considered approach achieves state of the art performances on benchmark large scale datasets.

Random walk kernels measure graph similarity by counting matching walks in two graphs. In their most popular form of geometric random walk kernels, longer walks of length $k$ are downweighted by a factor of $\lambda^k$ ($\lambda < 1$) to ensure convergence of the corresponding geometric series. We know from the field of link prediction that this downweighting often leads to a phenomenon referred to as halting: Longer walks are downweighted so much that the similarity score is completely dominated by the comparison of walks of length 1. This is a naive kernel between edges and vertices. We theoretically show that halting may occur in geometric random walk kernels. We also empirically quantify its impact in simulated datasets and popular graph classification benchmark datasets. Our findings promise to be instrumental in future graph kernel development and applications of random walk kernels.

Learning-from-crowds aims to design proper aggregation strategies to infer the unknown true labels from the noisy labels provided by ordinary web workers. This paper presents max-margin majority voting (M^3V) to improve the discriminative ability of majority voting and further presents a Bayesian generalization to incorporate the flexibility of generative methods on modeling noisy observations with worker confusion matrices. We formulate the joint learning as a regularized Bayesian inference problem, where the posterior regularization is derived by maximizing the margin between the aggregated score of a potential true label and that of any alternative label. Our Bayesian model naturally covers the Dawid-Skene estimator and M^3V. Empirical results demonstrate that our methods are competitive, often achieving better results than state-of-the-art estimators.

This paper establishes a statistical versus computational trade-offfor solving a basic high-dimensional machine learning problem via a basic convex relaxation method. Specifically, we consider the {\em Sparse Principal Component Analysis} (Sparse PCA) problem, and the family of {\em Sum-of-Squares} (SoS, aka Lasserre/Parillo) convex relaxations. It was well known that in large dimension $p$, a planted $k$-sparse unit vector can be {\em in principle} detected using only $n \approx k\log p$ (Gaussian or Bernoulli) samples, but all {\em efficient} (polynomial time) algorithms known require $n \approx k^2 $ samples. It was also known that this quadratic gap cannot be improved by the the most basic {\em semi-definite} (SDP, aka spectral) relaxation, equivalent to a degree-2 SoS algorithms. Here we prove that also degree-4 SoS algorithms cannot improve this quadratic gap. This average-case lower bound adds to the small collection of hardness results in machine learning for this powerful family of convex relaxation algorithms. Moreover, our design of moments (or ``pseudo-expectations'') for this lower bound is quite different than previous lower bounds. Establishing lower bounds for higher degree SoS algorithms for remains a challenging problem.

The process of dynamic state estimation (filtering) based on point process observations is in general intractable. Numerical sampling techniques are often practically useful, but lead to limited conceptual insight about optimal encoding/decoding strategies, which are of significant relevance to Computational Neuroscience. We develop an analytically tractable Bayesian approximation to optimal filtering based on point process observations, which allows us to introduce distributional assumptions about sensory cell properties, that greatly facilitates the analysis of optimal encoding in situations deviating from common assumptions of uniform coding. The analytic framework leads to insights which are difficult to obtain from numerical algorithms, and is consistent with experiments about the distribution of tuning curve centers. Interestingly, we find that the information gained from the absence of spikes may be crucial to performance.

Adaptive stochastic optimization optimizes an objective function adaptively under uncertainty. Adaptive stochastic optimization plays a crucial role in planning and learning under uncertainty, but is, unfortunately, computationally intractable in general. This paper introduces two conditions on the objective function, the marginal likelihood rate bound and the marginal likelihood bound, which enable efficient approximate solution of adaptive stochastic optimization. Several interesting classes of functions satisfy these conditions naturally, e.g., the version space reduction function for hypothesis learning. We describe Recursive Adaptive Coverage (RAC), a new adaptive stochastic optimization algorithm that exploits these conditions, and apply it to two planning tasks under uncertainty. In constrast to the earlier submodular optimization approach, our algorithm applies to adaptive stochastic optimization algorithm over both sets and paths.

Latent models are a fundamental modeling tool in machine learning applications, but they present significant computational and analytical challenges. The popular EM algorithm and its variants, is a much used algorithmic tool; yet our rigorous understanding of its performance is highly incomplete. Recently, work in [1] has demonstrated that for an important class of problems, EM exhibits linear local convergence. In the high-dimensional setting, however, the M-step may not be well defined. We address precisely this setting through a unified treatment using regularization. While regularization for high-dimensional problems is by now well understood, the iterative EM algorithm requires a careful balancing of making progress towards the solution while identifying the right structure (e.g., sparsity or low-rank). In particular, regularizing the M-step using the state-of-the-art high-dimensional prescriptions (e.g., \`a la [19]) is not guaranteed to provide this balance. Our algorithm and analysis are linked in a way that reveals the balance between optimization and statistical errors. We specialize our general framework to sparse gaussian mixture models, high-dimensional mixed regression, and regression with missing variables, obtaining statistical guarantees for each of these examples.

We analyze in this paper a random feature map based on a theory of invariance (I-theory) introduced in [1]. More specifically, a group invariant signal signature is obtained through cumulative distributions of group-transformed random projections. Ouranalysis bridges invariant feature learning with kernel methods, as we show that this feature map defines an expected Haar-integration kernel that is invariant to the specified group action. We show how this nonlinear random feature mapapproximates this group invariant kernel uniformly on a set of N points. Moreover, we show that it defines a function space that is dense in the equivalent Invariant Reproducing Kernel Hilbert Space. Finally, we quantify error rates of the convergence of the empirical risk minimization, as well as the reduction in the sample complexity of a learning algorithm using such an invariant representation for signal classification, in a classical supervised learning setting.