Latent factor models have been widely used to analyze simultaneous recordings of spike trains from large, heterogeneous neural populations. These models assume the signal of interest in the population is a low-dimensional latent intensity that evolves over time, which is observed in high dimension via noisy point-process observations. These techniques have been well used to capture neural correlations across a population and to provide a smooth, denoised, and concise representation of high-dimensional spiking data. One limitation of many current models is that the observation model is assumed to be Poisson, which lacks the flexibility to capture under-and over-dispersion that is common in recorded neural data, thereby introducing bias into estimates of covariance. Here we develop the generalized count linear dynamical system, which relaxes the Poisson assumption by using a more general exponential family for count data. In addition to containing Poisson, Bernoulli, negative binomial, and other common count distributions as special cases, we show that this model can be tractably learned by extending recent advances in variational inference techniques. We apply our model to data from primate motor cortex and demonstrate performance improvements over state-of-the-art methods, both in capturing the variance structure of the data and in held-out prediction.

Stochastic gradient descent (SGD) is a ubiquitous algorithm for a variety of machine learning problems. Researchers and industry have developed several techniques to optimize SGD's runtime performance, including asynchronous execution and reduced precision. Our main result is a martingale-based analysis that enables us to capture the rich noise models that may arise from such techniques.

Color constancy is the recovery of true surface color from observed color, and requires estimating the chromaticity of scene illumination to correct for the bias it induces. In this paper, we show that the per-pixel color statistics of natural scenes--without any spatial or semantic context--can by themselves be a powerful cue for color constancy. Specifically, we describe an illuminant estimation method that is built around a "classifier" for identifying the true chromaticity of a pixel given its luminance (absolute brightness across color channels). During inference, each pixel's observed color restricts its true chromaticity to those values that can be explained by one of a candidate set of illuminants, and applying the classifier over these values yields a distribution over the corresponding illuminants. A global estimate for the scene illuminant is computed through a simple aggregation of these distributions across all pixels.

We present a new algorithm for community detection. The algorithm uses random walks to embed the graph in a space of measures, after which a modification of k-means in that space is applied. The algorithm is therefore fast and easily parallelizable. We evaluate the algorithm on standard random graph benchmarks, including some overlapping community benchmarks, and find its performance to be better or at least as good as previously known algorithms.

The iteration complexity of the block-coordinate descent (BCD) type algorithm has been under extensive investigation. It was recently shown that for convex problems the classical cyclic BCGD (block coordinate gradient descent) achieves an O(1/r) complexity (r is the number of passes of all blocks). However, such bounds are at least linearly depend on K (the number of variable blocks), and are at least K times worse than those of the gradient descent (GD) and proximal gradient (PG) methods. In this paper, we close such theoretical performance gap between cyclic BCD and GD/PG.

We introduce the Gaussian Process Convolution Model (GPCM), a two-stage nonparametric generative procedure to model stationary signals as the convolution between a continuous-time white-noise process and a continuous-time linear filter drawn from Gaussian process. The GPCM is a continuous-time nonparametricwindow moving average process and, conditionally, is itself a Gaussian process with a nonparametric kernel defined in a probabilistic fashion. The generative model can be equivalently considered in the frequency domain, where the power spectral density of the signal is specified using a Gaussian process. One of the main contributions of the paper is to develop a novel variational freeenergy approach based on inter-domain inducing variables that efficiently learns the continuous-time linear filter and infers the driving white-noise process. In turn, this scheme provides closed-form probabilistic estimates of the covariance kernel and the noise-free signal both in denoising and prediction scenarios. Additionally, the variational inference procedure provides closed-form expressions for the approximate posterior of the spectral density given the observed data, leading to new Bayesian nonparametric approaches to spectrum estimation. The proposed GPCM is validated using synthetic and real-world signals.

Stochastic convex optimization is a basic and well studied primitive in machine learning. It is well known that convex and Lipschitz functions can be minimized efficiently using Stochastic Gradient Descent (SGD). The Normalized Gradient Descent (NGD) algorithm, is an adaptation of Gradient Descent, which updates according to the direction of the gradients, rather than the gradients themselves. In this paper we analyze a stochastic version of NGD and prove its convergence to a global minimum for a wider class of functions: we require the functions to be quasi-convex and locally-Lipschitz. Quasi-convexity broadens the concept of unimodality to multidimensions and allows for certain types of saddle points, which are a known hurdle for first-order optimization methods such as gradient descent. Locally-Lipschitz functions are only required to be Lipschitz in a small region around the optimum. This assumption circumvents gradient explosion, which is another known hurdle for gradient descent variants. Interestingly, unlike the vanilla SGD algorithm, the stochastic normalized gradient descent algorithm provably requires a minimal minibatch size.

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 highdimensional prescriptions (e.g., à 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.

In this work we aim at extending the theoretical foundations of lifelong learning. Previous work analyzing this scenario is based on the assumption that learning tasks are sampled i.i.d.

Clustering large data is a fundamental problem with a vast number of applications. Due to the increasing size of data, practitioners interested in clustering have turned to distributed computation methods. In this work, we consider the widely used k-center clustering problem and its variant used to handle noisy data, k-center with outliers. In the noise-free setting we demonstrate how a previously-proposed distributed method is actually an O(1)-approximation algorithm, which accurately explains its strong empirical performance. Additionally, in the noisy setting, we develop a novel distributed algorithm that is also an O(1)-approximation. These algorithms are highly parallel and lend themselves to virtually any distributed computing framework. We compare each empirically against the best known sequential clustering methods and show that both distributed algorithms are consistently close to their sequential versions. The algorithms are all one can hope for in distributed settings: they are fast, memory efficient and they match their sequential counterparts.