Low-rank matrix is desired in many machine learning and computer vision problems. Most of the recent studies use the nuclear norm as a convex surrogate of the rank operator. However, all singular values are simply added together by the nuclear norm, and thus the rank may not be well approximated in practical problems. In this paper, we propose to use a log-determinant (LogDet) function as a smooth and closer, though non-convex, approximation to rank for obtaining a low-rank representation in subspace clustering. Augmented Lagrange multipliers strategy is applied to iteratively optimize the LogDet-based non-convex objective function on potentially large-scale data. By making use of the angular information of principal directions of the resultant low-rank representation, an affinity graph matrix is constructed for spectral clustering. Experimental results on motion segmentation and face clustering data demonstrate that the proposed method often outperforms state-of-the-art subspace clustering algorithms.

This paper discusses the effect of hubness in zero-shot learning, when ridge regression is used to find a mapping between the example space to the label space. Contrary to the existing approach, which attempts to find a mapping from the example space to the label space, we show that mapping labels into the example space is desirable to suppress the emergence of hubs in the subsequent nearest neighbor search step. Assuming a simple data model, we prove that the proposed approach indeed reduces hubness. This was verified empirically on the tasks of bilingual lexicon extraction and image labeling: hubness was reduced with both of these tasks and the accuracy was improved accordingly.

We consider the problem of learning the inhomogeneous intensity of a counting process, under a sparse segmentation assumption. We introduce a weighted total-variation penalization, using data-driven weights that correctly scale the penalization along the observation interval. We prove that this leads to a sharp tuning of the convex relaxation of the segmentation prior, by stating oracle inequalities with fast rates of convergence, and consistency for change-points detection. This provides first theoretical guarantees for segmentation with a convex proxy beyond the standard i.i.d signal + white noise setting. We introduce a fast algorithm to solve this convex problem. Numerical experiments illustrate our approach on simulated and on a high-frequency genomics dataset.

This paper compares classical parametric methods with recently developed Bayesian methods for system identification. A Full Bayes solution is considered together with one of the standard approximations based on the Empirical Bayes paradigm. Results regarding point estimators for the impulse response as well as for confidence regions are reported.

A new Bayesian approach to linear system identification has been proposed in a series of recent papers. The main idea is to frame linear system identification as predictor estimation in an infinite dimensional space, with the aid of regularization/Bayesian techniques. This approach guarantees the identification of stable predictors based on the prediction error minimization. Unluckily, the stability of the predictors does not guarantee the stability of the impulse response of the system. In this paper we propose and compare various techniques to address this issue. Simulations results comparing these techniques will be provided.

We introduce a novel algorithm for solving learning problems where both the loss function and the regularizer are non-convex but belong to the class of difference of convex (DC) functions. Our contribution is a new general purpose proximal Newton algorithm that is able to deal with such a situation. The algorithm consists in obtaining a descent direction from an approximation of the loss function and then in performing a line search to ensure sufficient descent. A theoretical analysis is provided showing that the iterates of the proposed algorithm {admit} as limit points stationary points of the DC objective function. Numerical experiments show that our approach is more efficient than current state of the art for a problem with a convex loss functions and non-convex regularizer. We have also illustrated the benefit of our algorithm in high-dimensional transductive learning problem where both loss function and regularizers are non-convex.

We introduce a very general method for sparse and large-scale variable selection. The large-scale regression settings is such that both the number of parameters and the number of samples are extremely large. The proposed method is based on careful combination of penalized estimators, each applied to a random projection of the sample space into a low-dimensional space. In one special case that we study in detail, the random projections are divided into non-overlapping blocks; each consisting of only a small portion of the original data. Within each block we select the projection yielding the smallest out-of-sample error. Our random ensemble estimator then aggregates the results according to new maximal-contrast voting scheme to determine the final selected set. Our theoretical results illuminate the effect on performance of increasing the number of non-overlapping blocks. Moreover, we demonstrate that statistical optimality is retained along with the computational speedup. The proposed method achieves minimax rates for approximate recovery over all estimators using the full set of samples. Furthermore, our theoretical results allow the number of subsamples to grow with the subsample size and do not require irrepresentable condition. The estimator is also compared empirically with several other popular high-dimensional estimators via an extensive simulation study, which reveals its excellent finite-sample performance.

In this paper, we build an organization of high-dimensional datasets that cannot be cleanly embedded into a low-dimensional representation due to missing entries and a subset of the features being irrelevant to modeling functions of interest. Our algorithm begins by defining coarse neighborhoods of the points and defining an expected empirical function value on these neighborhoods. We then generate new non-linear features with deep net representations tuned to model the approximate function, and re-organize the geometry of the points with respect to the new representation. Finally, the points are locally z-scored to create an intrinsic geometric organization which is independent of the parameters of the deep net, a geometry designed to assure smoothness with respect to the empirical function. We examine this approach on data from the Center for Medicare and Medicaid Services Hospital Quality Initiative, and generate an intrinsic low-dimensional organization of the hospitals that is smooth with respect to an expert driven function of quality.

We consider the problem of online nonparametric regression with arbitrary deterministic sequences. Using ideas from the chaining technique, we design an algorithm that achieves a Dudley-type regret bound similar to the one obtained in a non-constructive fashion by Rakhlin and Sridharan (2014). Our regret bound is expressed in terms of the metric entropy in the sup norm, which yields optimal guarantees when the metric and sequential entropies are of the same order of magnitude. In particular our algorithm is the first one that achieves optimal rates for online regression over H{\"o}lder balls. In addition we show for this example how to adapt our chaining algorithm to get a reasonable computational efficiency with similar regret guarantees (up to a log factor).

Many applications require recovering a matrix of minimal rank within an affine constraint set, with matrix completion a notable special case. Because the problem is NP-hard in general, it is common to replace the matrix rank with the nuclear norm, which acts as a convenient convex surrogate. While elegant theoretical conditions elucidate when this replacement is likely to be successful, they are highly restrictive and convex algorithms fail when the ambient rank is too high or when the constraint set is poorly structured. Non-convex alternatives fare somewhat better when carefully tuned; however, convergence to locally optimal solutions remains a continuing source of failure. Against this backdrop we derive a deceptively simple and parameter-free probabilistic PCA-like algorithm that is capable, over a wide battery of empirical tests, of successful recovery even at the theoretical limit where the number of measurements equal the degrees of freedom in the unknown low-rank matrix. Somewhat surprisingly, this is possible even when the affine constraint set is highly ill-conditioned. While proving general recovery guarantees remains evasive for non-convex algorithms, Bayesian-inspired or otherwise, we nonetheless show conditions whereby the underlying cost function has a unique stationary point located at the global optimum; no existing cost function we are aware of satisfies this same property. We conclude with a simple computer vision application involving image rectification and a standard collaborative filtering benchmark.