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Kernel ridge regression (KRR) is a popular scheme for non-linear non-parametric learning. However, existing implementations of KRR require that all the data is stored in the main memory, which severely limits the use of KRR in contexts where data size far exceeds the memory size. Such applications are increasingly common in data mining, bioinformatics, and control. A powerful paradigm for computing on data sets that are too large for memory is the streaming model of computation, where we process one data sample at a time, discarding each sample before moving on to the next one. In this paper, we propose StreaMRAK - a streaming version of KRR. StreaMRAK improves on existing KRR schemes by dividing the problem into several levels of resolution, which allows continual refinement to the predictions. The algorithm reduces the memory requirement by continuously and efficiently integrating new samples into the training model. With a novel sub-sampling scheme, StreaMRAK reduces memory and computational complexities by creating a sketch of the original data, where the sub-sampling density is adapted to the bandwidth of the kernel and the local dimensionality of the data. We present a showcase study on two synthetic problems and the prediction of the trajectory of a double pendulum. The results show that the proposed algorithm is fast and accurate.

We introduce a construction of multiscale tight frames on general domains. The frame elements are obtained by spectral filtering of the integral operator associated with a reproducing kernel. Our construction extends classical wavelets as well as generalized wavelets on both continuous and discrete non-Euclidean structures such as Riemannian manifolds and weighted graphs. Moreover, it allows to study the relation between continuous and discrete frames in a random sampling regime, where discrete frames can be seen as Monte Carlo estimates of the continuous ones. Pairing spectral regularization with learning theory, we show that a sample frame tends to its population counterpart, and derive explicit finite-sample rates on spaces of Sobolev and Besov regularity. Our results prove the stability of frames constructed on empirical data, in the sense that all stochastic discretizations have the same underlying limit regardless of the set of initial training samples.

In this work we consider numerical efficiency and convergence rates for solvers of non-convex multi-penalty formulations when reconstructing sparse signals from noisy linear measurements. We extend an existing approach, based on reduction to an augmented single-penalty formulation, to the non-convex setting and discuss its computational intractability in large-scale applications. To circumvent this limitation, we propose an alternative single-penalty reduction based on infimal convolution that shares the benefits of the augmented approach but is computationally less dependent on the problem size. We provide linear convergence rates for both approaches, and their dependence on design parameters. Numerical experiments substantiate our theoretical findings., Comment: 20 pages, 2 figures

In this paper we propose and study a family of continuous wavelets on general domains, and a corresponding stochastic discretization that we call Monte Carlo wavelets. First, using tools from the theory of reproducing kernel Hilbert spaces and associated integral operators, we define a family of continuous wavelets by spectral calculus. Then, we propose a stochastic discretization based on Monte Carlo estimates of integral operators. Using concentration of measure results, we establish the convergence of such a discretization and derive convergence rates under natural regularity assumptions.

Single index model is a powerful yet simple model, widely used in statistics, machine learning, and other scientific fields. It models the regression function as $g()$, where a is an unknown index vector and x are the features. This paper deals with a nonlinear generalization of this framework to allow for a regressor that uses multiple index vectors, adapting to local changes in the responses. To do so we exploit the conditional distribution over function-driven partitions, and use linear regression to locally estimate index vectors. We then regress by applying a kNN type estimator that uses a localized proxy of the geodesic metric. We present theoretical guarantees for estimation of local index vectors and out-of-sample prediction, and demonstrate the performance of our method with experiments on synthetic and real-world data sets, comparing it with state-of-the-art methods., Comment: 37 pages, 23 figures, 4 table

We consider the problem of recovering an unknown effectively $(s_1,s_2)$-sparse low-rank-$R$ matrix $X$ with possibly non-orthogonal rank-$1$ decomposition from incomplete and inaccurate linear measurements of the form $y = \mathcal A (X) + \eta$, where $\eta$ is an ineliminable noise. We first derive an optimization formulation for matrix recovery under the considered model and propose a novel algorithm, called Alternating Tikhonov regularization and Lasso (A-T-LA$\text{S}_{2,1}$), to solve it. The algorithm is based on a multi-penalty regularization, which is able to leverage both structures (low-rankness and sparsity) simultaneously. The algorithm is a fast first order method, and straightforward to implement. We prove global convergence for any linear measurement model to stationary points and local convergence to global minimizers. By adapting the concept of restricted isometry property from compressed sensing to our novel model class, we prove error bounds between global minimizers and ground truth, up to noise level, from a number of subgaussian measurements scaling as $R(s_1+s_2)$, up to log-factors in the dimension, and relative-to-diameter distortion. Simulation results demonstrate both the accuracy and efficacy of the algorithm, as well as its superiority to the state-of-the-art algorithms in strong noise regimes and for matrices, whose singular vectors do not possess exact (joint-) sparse support.

For many algorithms, parameter tuning remains a challenging and critical task, which becomes tedious and infeasible in a multi-parameter setting. Multi-penalty regularization, successfully used for solving undetermined sparse regression of problems of unmixing type where signal and noise are additively mixed, is one of such examples. In this paper, we propose a novel algorithmic framework for an adaptive parameter choice in multi-penalty regularization with a focus on the correct support recovery. Building upon the theory of regularization paths and algorithms for single-penalty functionals, we extend these ideas to a multi-penalty framework by providing an efficient procedure for the construction of regions containing structurally similar solutions, i.e., solutions with the same sparsity and sign pattern, over the whole range of parameters. Combining this with a model selection criterion, we can choose regularization parameters in a data-adaptive manner. Another advantage of our algorithm is that it provides an overview on the solution stability over the whole range of parameters. This can be further exploited to obtain additional insights into the problem of interest. We provide a numerical analysis of our method and compare it to the state-of-the-art single-penalty algorithms for compressed sensing problems in order to demonstrate the robustness and power of the proposed algorithm.

This paper extends the recently proposed and theoretically justified iterative thresholding and $K$ residual means algorithm ITKrM to learning dicionaries from incomplete/masked training data (ITKrMM). It further adapts the algorithm to the presence of a low rank component in the data and provides a strategy for recovering this low rank component again from incomplete data. Several synthetic experiments show the advantages of incorporating information about the corruption into the algorithm. Finally, image inpainting is considered as application example, which demonstrates the superior performance of ITKrMM in terms of speed at similar or better reconstruction quality compared to its closest dictionary learning counterpart., Comment: 22 pages, 9 figures, (this version with bug fix for wksvd)

Despite a variety of available techniques the issue of the proper regularization parameter choice for inverse problems still remains one of the biggest challenges. The main difficulty lies in constructing a rule, allowing to compute the parameter from given noisy data without relying either on a priori knowledge of the solution or on the noise level. In this paper we propose a novel method based on supervised machine learning to approximate the high-dimensional function, mapping noisy data into a good approximation to the optimal Tikhonov regularization parameter. Our assumptions are that solutions of the inverse problem are statistically distributed in a concentrated manner on (lower-dimensional) linear subspaces and the noise is sub-gaussian. One of the surprising facts is that the number of previously observed examples for the supervised learning of the optimal parameter mapping scales at most linearly with the dimension of the solution subspace. We also provide explicit error bounds on the accuracy of the approximated parameter and the corresponding regularization solution. Even though the results are more of theoretical nature, we present a recipe for the practical implementation of the approach and provide numerical experiments confirming the theoretical results. We also outline interesting directions for future research with some preliminary results, confirming their feasibility.

Inspired by several real-life applications in audio processing and medical image analysis, where the quantity of interest is generated by several sources to be accurately modeled and separated, as well as by recent advances in regularization theory and optimization, we study the conditions on optimal support recovery in inverse problems of unmixing type by means of multi-penalty regularization. We consider and analyze a regularization functional composed of a data-fidelity term, where signal and noise are additively mixed, a non-smooth, convex, sparsity promoting term, and a quadratic penalty term to model the noise. We prove not only that the well-established theory for sparse recovery in the single parameter case can be translated to the multi-penalty settings, but we also demonstrate the enhanced properties of multi-penalty regularization in terms of support identification compared to sole $\ell^1$-minimization. We additionally confirm and support the theoretical results by extensive numerical simulations, which give a statistics of robustness of the multi-penalty regularization scheme with respect to the single-parameter counterpart. Eventually, we confirm a significant improvement in performance compared to standard $\ell^1$-regularization for compressive sensing problems considered in our experiments.

Inspired by several recent developments in regularization theory, optimization, and signal processing, we present and analyze a numerical approach to multi-penalty regularization in spaces of sparsely represented functions. The sparsity prior is motivated by the largely expected geometrical/structured features of high-dimensional data, which may not be well-represented in the framework of typically more isotropic Hilbert spaces. In this paper, we are particularly interested in regularizers which are able to correctly model and separate the multiple components of additively mixed signals. This situation is rather common as pure signals may be corrupted by additive noise. To this end, we consider a regularization functional composed by a data-fidelity term, where signal and noise are additively mixed, a non-smooth and non-convex sparsity promoting term, and a penalty term to model the noise. We propose and analyze the convergence of an iterative alternating algorithm based on simple iterative thresholding steps to perform the minimization of the functional. By means of this algorithm, we explore the effect of choosing different regularization parameters and penalization norms in terms of the quality of recovering the pure signal and separating it from additive noise. For a given fixed noise level numerical experiments confirm a significant improvement in performance compared to standard one-parameter regularization methods. By using high-dimensional data analysis methods such as Principal Component Analysis, we are able to show the correct geometrical clustering of regularized solutions around the expected solution. Eventually, for the compressive sensing problems considered in our experiments we provide a guideline for a choice of regularization norms and parameters., Comment: 32 pages