Mechanisms of human color vision are characterized by two phenomenological aspects: the system is nonlinear and adaptive to changing environments. Conventional attempts to derive these features from statistics use separate arguments for each aspect. The few statistical approaches that do consider both phenomena simultaneously follow parametric formulations based on empirical models. Therefore, it may be argued that the behavior does not come directly from the color statistics but from the convenient functional form adopted. In addition, many times the whole statistical analysis is based on simplified databases that disregard relevant physical effects in the input signal, as for instance by assuming flat Lambertian surfaces. Here we address the simultaneous statistical explanation of (i) the nonlinear behavior of achromatic and chromatic mechanisms in a fixed adaptation state, and (ii) the change of such behavior. Both phenomena emerge directly from the samples through a single data-driven method: the Sequential Principal Curves Analysis (SPCA) with local metric. SPCA is a new manifold learning technique to derive a set of sensors adapted to the manifold using different optimality criteria. A new database of colorimetrically calibrated images of natural objects under these illuminants was collected. The results obtained by applying SPCA show that the psychophysical behavior on color discrimination thresholds, discount of the illuminant and corresponding pairs in asymmetric color matching, emerge directly from realistic data regularities assuming no a priori functional form. These results provide stronger evidence for the hypothesis of a statistically driven organization of color sensors. Moreover, the obtained results suggest that color perception at this low abstraction level may be guided by an error minimization strategy rather than by the information maximization principle.

This paper presents a new framework for manifold learning based on a sequence of principal polynomials that capture the possibly nonlinear nature of the data. The proposed Principal Polynomial Analysis (PPA) generalizes PCA by modeling the directions of maximal variance by means of curves, instead of straight lines. Contrarily to previous approaches, PPA reduces to performing simple univariate regressions, which makes it computationally feasible and robust. Moreover, PPA shows a number of interesting analytical properties. First, PPA is a volume-preserving map, which in turn guarantees the existence of the inverse. Second, such an inverse can be obtained in closed form. Invertibility is an important advantage over other learning methods, because it permits to understand the identified features in the input domain where the data has physical meaning. Moreover, it allows to evaluate the performance of dimensionality reduction in sensible (input-domain) units. Volume preservation also allows an easy computation of information theoretic quantities, such as the reduction in multi-information after the transform. Third, the analytical nature of PPA leads to a clear geometrical interpretation of the manifold: it allows the computation of Frenet-Serret frames (local features) and of generalized curvatures at any point of the space. And fourth, the analytical Jacobian allows the computation of the metric induced by the data, thus generalizing the Mahalanobis distance. These properties are demonstrated theoretically and illustrated experimentally. The performance of PPA is evaluated in dimensionality and redundancy reduction, in both synthetic and real datasets from the UCI repository.

In this work we propose a new deep learning tool called deep dictionary learning. Multi-level dictionaries are learnt in a greedy fashion, one layer at a time. This requires solving a simple (shallow) dictionary learning problem, the solution to this is well known. We apply the proposed technique on some benchmark deep learning datasets. We compare our results with other deep learning tools like stacked autoencoder and deep belief network; and state of the art supervised dictionary learning tools like discriminative KSVD and label consistent KSVD. Our method yields better results than all.

The need for new methods to deal with big data is a common theme in most scientific fields, although its definition tends to vary with the context. Statistical ideas are an essential part of this, and as a partial response, a thematic program on statistical inference, learning, and models in big data was held in 2015 in Canada, under the general direction of the Canadian Statistical Sciences Institute, with major funding from, and most activities located at, the Fields Institute for Research in Mathematical Sciences. This paper gives an overview of the topics covered, describing challenges and strategies that seem common to many different areas of application, and including some examples of applications to make these challenges and strategies more concrete.

While the harmonic function solution performs well in many semi-supervised learning (SSL) tasks, it is known to scale poorly with the number of samples. Recent successful and scalable methods, such as the eigenfunction method focus on efficiently approximating the whole spectrum of the graph Laplacian constructed from the data. This is in contrast to various subsampling and quantization methods proposed in the past, which may fail in preserving the graph spectra. However, the impact of the approximation of the spectrum on the final generalization error is either unknown, or requires strong assumptions on the data. In this paper, we introduce Sparse-HFS, an efficient edge-sparsification algorithm for SSL. By constructing an edge-sparse and spectrally similar graph, we are able to leverage the approximation guarantees of spectral sparsification methods to bound the generalization error of Sparse-HFS. As a result, we obtain a theoretically-grounded approximation scheme for graph-based SSL that also empirically matches the performance of known large-scale methods.

High-velocity streams of high-dimensional data pose significant "big data" analysis challenges across a range of applications and settings. Online learning and online convex programming play a significant role in the rapid recovery of important or anomalous information from these large datastreams. While recent advances in online learning have led to novel and rapidly converging algorithms, these methods are unable to adapt to nonstationary environments arising in real-world problems. This paper describes a dynamic mirror descent framework which addresses this challenge, yielding low theoretical regret bounds and accurate, adaptive, and computationally efficient algorithms which are applicable to broad classes of problems. The methods are capable of learning and adapting to an underlying and possibly time-varying dynamical model. Empirical results in the context of dynamic texture analysis, solar flare detection, sequential compressed sensing of a dynamic scene, traffic surveillance,tracking self-exciting point processes and network behavior in the Enron email corpus support the core theoretical findings.

Deep convolutional networks provide state of the art classifications and regressions results over many high-dimensional problems. We review their architecture, which scatters data with a cascade of linear filter weights and non-linearities. A mathematical framework is introduced to analyze their properties. Computations of invariants involve multiscale contractions, the linearization of hierarchical symmetries, and sparse separations. Applications are discussed.

Asynchronous distributed stochastic gradient descent methods have trouble converging because of stale gradients. A gradient update sent to a parameter server by a client is stale if the parameters used to calculate that gradient have since been updated on the server. Approaches have been proposed to circumvent this problem that quantify staleness in terms of the number of elapsed updates. In this work, we propose a novel method that quantifies staleness in terms of moving averages of gradient statistics. We show that this method outperforms previous methods with respect to convergence speed and scalability to many clients. We also discuss how an extension to this method can be used to dramatically reduce bandwidth costs in a distributed training context. In particular, our method allows reduction of total bandwidth usage by a factor of 5 with little impact on cost convergence. We also describe (and link to) a software library that we have used to simulate these algorithms deterministically on a single machine.

Slow feature analysis (SFA) is an unsupervised-learning algorithm that extracts slowly varying features from a multi-dimensional time series. A supervised extension to SFA for classification and regression is graph-based SFA (GSFA). GSFA is based on the preservation of similarities, which are specified by a graph structure derived from the labels. It has been shown that hierarchical GSFA (HGSFA) allows learning from images and other high-dimensional data. The feature space spanned by HGSFA is complex due to the composition of the nonlinearities of the nodes in the network. However, we show that the network discards useful information prematurely before it reaches higher nodes, resulting in suboptimal global slowness and an under-exploited feature space. To counteract these problems, we propose an extension called hierarchical information-preserving GSFA (HiGSFA), where information preservation complements the slowness-maximization goal. We build a 10-layer HiGSFA network to estimate human age from facial photographs of the MORPH-II database, achieving a mean absolute error of 3.50 years, improving the state-of-the-art performance. HiGSFA and HGSFA support multiple-labels and offer a rich feature space, feed-forward training, and linear complexity in the number of samples and dimensions. Furthermore, HiGSFA outperforms HGSFA in terms of feature slowness, estimation accuracy and input reconstruction, giving rise to a promising hierarchical supervised-learning approach.

In this paper, we propose an active perception method for recognizing object categories based on the multimodal hierarchical Dirichlet process (MHDP). The MHDP enables a robot to form object categories using multimodal information, e.g., visual, auditory, and haptic information, which can be observed by performing actions on an object. However, performing many actions on a target object requires a long time. In a real-time scenario, i.e., when the time is limited, the robot has to determine the set of actions that is most effective for recognizing a target object. We propose an MHDP-based active perception method that uses the information gain (IG) maximization criterion and lazy greedy algorithm. We show that the IG maximization criterion is optimal in the sense that the criterion is equivalent to a minimization of the expected Kullback--Leibler divergence between a final recognition state and the recognition state after the next set of actions. However, a straightforward calculation of IG is practically impossible. Therefore, we derive an efficient Monte Carlo approximation method for IG by making use of a property of the MHDP. We also show that the IG has submodular and non-decreasing properties as a set function because of the structure of the graphical model of the MHDP. Therefore, the IG maximization problem is reduced to a submodular maximization problem. This means that greedy and lazy greedy algorithms are effective and have a theoretical justification for their performance. We conducted an experiment using an upper-torso humanoid robot and a second one using synthetic data. The experimental results show that the method enables the robot to select a set of actions that allow it to recognize target objects quickly and accurately. The results support our theoretical outcomes.