arXiv subjects

Konstantinos Slavakis

Publications and source records attributed to Konstantinos Slavakis.

At least 37 records · Page 2Linked to original sources

arXiv2024

Multilinear Kernel Regression and Imputation via Manifold Learning

This paper introduces a novel nonparametric framework for data imputation, coined multilinear kernel regression and imputation via the manifold assumption (MultiL-KRIM). Motivated by manifold learning, MultiL-KRIM models data features as a point cloud located in or close to a user-unknown smooth manifold embedded in a reproducing kernel Hilbert space. Unlike typical manifold-learning routes, which seek low-dimensional patterns via regularizers based on graph-Laplacian matrices, MultiL-KRIM builds instead on the intuitive concept of tangent spaces to manifolds and incorporates collaboration among point-cloud neighbors (regressors) directly into the data-modeling term of the loss function. Multiple kernel functions are allowed to offer robustness and rich approximation properties, while multiple matrix factors offer low-rank modeling, integrate dimensionality reduction, and streamline computations with no need of training data. Two important application domains showcase the functionality of MultiL-KRIM: time-varying-graph-signal (TVGS) recovery, and reconstruction of highly accelerated dynamic-magnetic-resonance-imaging (dMRI) data. Extensive numerical tests on real and synthetic data demonstrate MultiL-KRIM's remarkable speedups over its predecessors, and outperformance over prevalent "shallow" data-imputation techniques, with a more intuitive and explainable pipeline than deep-image-prior methods.

eess.SP↗

arXiv2023

Proximal Bellman mappings for reinforcement learning and their application to robust adaptive filtering

This paper aims at the algorithmic/theoretical core of reinforcement learning (RL) by introducing the novel class of proximal Bellman mappings. These mappings are defined in reproducing kernel Hilbert spaces (RKHSs), to benefit from the rich approximation properties and inner product of RKHSs, they are shown to belong to the powerful Hilbertian family of (firmly) nonexpansive mappings, regardless of the values of their discount factors, and possess ample degrees of design freedom to even reproduce attributes of the classical Bellman mappings and to pave the way for novel RL designs. An approximate policy-iteration scheme is built on the proposed class of mappings to solve the problem of selecting online, at every time instance, the "optimal" exponent $p$ in a $p$-norm loss to combat outliers in linear adaptive filtering, without training data and any knowledge on the statistical properties of the outliers. Numerical tests on synthetic data showcase the superior performance of the proposed framework over several non-RL and kernel-based RL schemes.

eess.SP↗

arXiv2023

Multi-Linear Kernel Regression and Imputation in Data Manifolds

This paper introduces an efficient multi-linear nonparametric (kernel-based) approximation framework for data regression and imputation, and its application to dynamic magnetic-resonance imaging (dMRI). Data features are assumed to reside in or close to a smooth manifold embedded in a reproducing kernel Hilbert space. Landmark points are identified to describe concisely the point cloud of features by linear approximating patches which mimic the concept of tangent spaces to smooth manifolds. The multi-linear model effects dimensionality reduction, enables efficient computations, and extracts data patterns and their geometry without any training data or additional information. Numerical tests on dMRI data under severe under-sampling demonstrate remarkable improvements in efficiency and accuracy of the proposed approach over its predecessors, popular data modeling methods, as well as recent tensor-based and deep-image-prior schemes.

eess.SP↗

arXiv2022

Dynamic selection of p-norm in linear adaptive filtering via online kernel-based reinforcement learning

This study addresses the problem of selecting dynamically, at each time instance, the ``optimal'' p-norm to combat outliers in linear adaptive filtering without any knowledge on the potentially time-varying probability distribution function of the outliers. To this end, an online and data-driven framework is designed via kernel-based reinforcement learning (KBRL). Novel Bellman mappings on reproducing kernel Hilbert spaces (RKHSs) are introduced that need no knowledge on transition probabilities of Markov decision processes, and are nonexpansive with respect to the underlying Hilbertian norm. An approximate policy-iteration framework is finally offered via the introduction of a finite-dimensional affine superset of the fixed-point set of the proposed Bellman mappings. The well-known ``curse of dimensionality'' in RKHSs is addressed by building a basis of vectors via an approximate linear dependency criterion. Numerical tests on synthetic data demonstrate that the proposed framework selects always the ``optimal'' p-norm for the outlier scenario at hand, outperforming at the same time several non-RL and KBRL schemes.

eess.SP↗

arXiv2022

online and lightweight kernel-based approximated policy iteration for dynamic p-norm linear adaptive filtering

This paper introduces a solution to the problem of selecting dynamically (online) the ``optimal'' p-norm to combat outliers in linear adaptive filtering without any knowledge on the probability density function of the outliers. The proposed online and data-driven framework is built on kernel-based reinforcement learning (KBRL). To this end, novel Bellman mappings on reproducing kernel Hilbert spaces (RKHSs) are introduced. These mappings do not require any knowledge on transition probabilities of Markov decision processes, and are nonexpansive with respect to the underlying Hilbertian norm. The fixed-point sets of the proposed Bellman mappings are utilized to build an approximate policy-iteration (API) framework for the problem at hand. To address the ``curse of dimensionality'' in RKHSs, random Fourier features are utilized to bound the computational complexity of the API. Numerical tests on synthetic data for several outlier scenarios demonstrate the superior performance of the proposed API framework over several non-RL and KBRL schemes.

cs.LG↗

arXiv2020

Kernel Bi-Linear Modeling for Reconstructing Data on Manifolds: The Dynamic-MRI Case

This paper establishes a kernel-based framework for reconstructing data on manifolds, tailored to fit the dynamic-(d)MRI-data recovery problem. The proposed methodology exploits simple tangent-space geometries of manifolds in reproducing kernel Hilbert spaces and follows classical kernel-approximation arguments to form the data-recovery task as a bi-linear inverse problem. Departing from mainstream approaches, the proposed methodology uses no training data, employs no graph Laplacian matrix to penalize the optimization task, uses no costly (kernel) pre-imaging step to map feature points back to the input space, and utilizes complex-valued kernel functions to account for k-space data. The framework is validated on synthetically generated dMRI data, where comparisons against state-of-the-art schemes highlight the rich potential of the proposed approach in data-recovery problems.

cs.LG↗

arXiv2020

Network Clustering Via Kernel-ARMA Modeling and the Grassmannian The Brain-Network Case

This paper introduces a clustering framework for networks with nodes annotated with time-series data. The framework addresses all types of network-clustering problems: State clustering, node clustering within states (a.k.a. topology identification or community detection), and even subnetwork-state-sequence identification/tracking. Via a bottom-up approach, features are first extracted from the raw nodal time-series data by kernel autoregressive-moving-average modeling to reveal non-linear dependencies and low-rank representations, and then mapped onto the Grassmann manifold (Grassmannian). All clustering tasks are performed by leveraging the underlying Riemannian geometry of the Grassmannian in a novel way. To validate the proposed framework, brain-network clustering is considered, where extensive numerical tests on synthetic and real functional magnetic resonance imaging (fMRI) data demonstrate that the advocated learning framework compares favorably versus several state-of-the-art clustering schemes.

cs.LG↗

arXiv2019

Robust Hierarchical-Optimization RLS Against Sparse Outliers

This paper fortifies the recently introduced hierarchical-optimization recursive least squares (HO-RLS) against outliers which contaminate infrequently linear-regression models. Outliers are modeled as nuisance variables and are estimated together with the linear filter/system variables via a sparsity-inducing (non-)convexly regularized least-squares task. The proposed outlier-robust HO-RLS builds on steepest-descent directions with a constant step size (learning rate), needs no matrix inversion (lemma), accommodates colored nominal noise of known correlation matrix, exhibits small computational footprint, and offers theoretical guarantees, in a probabilistic sense, for the convergence of the system estimates to the solutions of a hierarchical-optimization problem: Minimize a convex loss, which models a-priori knowledge about the unknown system, over the minimizers of the classical ensemble LS loss. Extensive numerical tests on synthetically generated data in both stationary and non-stationary scenarios showcase notable improvements of the proposed scheme over state-of-the-art techniques.

cs.LG↗

arXiv2019

Bi-Linear Modeling of Data Manifolds for Dynamic-MRI Recovery

This paper puts forth a novel bi-linear modeling framework for data recovery via manifold-learning and sparse-approximation arguments and considers its application to dynamic magnetic-resonance imaging (dMRI). Each temporal-domain MR image is viewed as a point that lies onto or close to a smooth manifold, and landmark points are identified to describe the point cloud concisely. To facilitate computations, a dimensionality reduction module generates low-dimensional/compressed renditions of the landmark points. Recovery of the high-fidelity MRI data is realized by solving a non-convex minimization task for the linear decompression operator and those affine combinations of landmark points which locally approximate the latent manifold geometry. An algorithm with guaranteed convergence to stationary solutions of the non-convex minimization task is also provided. The aforementioned framework exploits the underlying spatio-temporal patterns and geometry of the acquired data without any prior training on external data or information. Extensive numerical results on simulated as well as real cardiac-cine and perfusion MRI data illustrate noteworthy improvements of the advocated machine-learning framework over state-of-the-art reconstruction techniques.

eess.IV↗

arXiv2019

Brain-Network Clustering via Kernel-ARMA Modeling and the Grassmannian

Recent advances in neuroscience and in the technology of functional magnetic resonance imaging (fMRI) and electro-encephalography (EEG) have propelled a growing interest in brain-network clustering via time-series analysis. Notwithstanding, most of the brain-network clustering methods revolve around state clustering and/or node clustering (a.k.a. community detection or topology inference) within states. This work answers first the need of capturing non-linear nodal dependencies by bringing forth a novel feature-extraction mechanism via kernel autoregressive-moving-average modeling. The extracted features are mapped to the Grassmann manifold (Grassmannian), which consists of all linear subspaces of a fixed rank. By virtue of the Riemannian geometry of the Grassmannian, a unifying clustering framework is offered to tackle all possible clustering problems in a network: Cluster multiple states, detect communities within states, and even identify/track subnetwork state sequences. The effectiveness of the proposed approach is underlined by extensive numerical tests on synthetic and real fMRI/EEG data which demonstrate that the advocated learning method compares favorably versus several state-of-the-art clustering schemes.

cs.LG↗

arXiv2019

The Stochastic Fejér-Monotone Hybrid Steepest Descent Method and the Hierarchical RLS

This paper introduces the stochastic Fejér-monotone hybrid steepest descent method (S-FM-HSDM) to solve affinely constrained and composite convex minimization tasks. The minimization task is not known exactly; noise contaminates the information about the composite loss function and the affine constraints. S-FM-HSDM generates sequences of random variables that, under certain conditions and with respect to a probability space, converge point-wise to solutions of the noiseless minimization task. S-FM-HSDM enjoys desirable attributes of optimization techniques such as splitting of variables and constant step size (learning rate). Furthermore, it provides a novel way of exploiting the information about the affine constraints via fixed-point sets of appropriate nonexpansive mappings. Among the offsprings of S-FM-HSDM, the hierarchical recursive least squares (HRLS) takes advantage of S-FM-HSDM's versatility toward affine constraints and offers a novel twist to LS by generating sequences of estimates that converge to solutions of a hierarchical optimization task: Minimize a convex loss over the set of minimizers of the ensemble LS loss. Numerical tests on a sparsity-aware LS task show that HRLS compares favorably to several state-of-the-art convex, as well as non-convex, stochastic-approximation and online-learning counterparts.

math.OC↗

arXiv2018

Fejér-monotone hybrid steepest descent method for affinely constrained and composite convex minimization tasks

This paper introduces the Fejér-monotone hybrid steepest descent method (FM-HSDM), a new member to the HSDM family of algorithms, for solving affinely constrained minimization tasks in real Hilbert spaces, where convex smooth and non-smooth losses compose the objective function. FM-HSDM offers sequences of estimates which converge weakly and, under certain hypotheses, strongly to solutions of the task at hand. Fixed-point theory, variational inequalities and affine-nonexpansive mappings are utilized to devise a scheme that accommodates affine constraints in a more versatile way than state-of-the-art primal-dual techniques and the alternating direction method of multipliers do. Recursions can be tuned to score low computational footprints, well-suited for large-scale optimization tasks, without compromising convergence guarantees. In contrast to its HSDM's precursors, FM-HSDM enjoys Fejér monotonicity, the step-size parameter stays constant across iterations to promote convergence speed-ups of the sequence of estimates to a minimizer, while only Lipschitzian continuity, and not strong monotonicity, of the derivative of the smooth-loss function is needed to ensure convergence. Results on the rate of convergence to an optimal point are also presented.

math.OC↗

arXiv2017

Per-Block-Convex Data Modeling by Accelerated Stochastic Approximation

Applications involving dictionary learning, non-negative matrix factorization, subspace clustering, and parallel factor tensor decomposition tasks motivate well algorithms for per-block-convex and non-smooth optimization problems. By leveraging the stochastic approximation paradigm and first-order acceleration schemes, this paper develops an online and modular learning algorithm for a large class of non-convex data models, where convexity is manifested only per-block of variables whenever the rest of them are held fixed. The advocated algorithm incurs computational complexity that scales linearly with the number of unknowns. Under minimal assumptions on the cost functions of the composite optimization task, without bounding constraints on the optimization variables, or any explicit information on bounds of Lipschitz coefficients, the expected cost evaluated online at the resultant iterates is provably convergent with quadratic rate to an accumulation point of the (per-block) minima, while subgradients of the expected cost asymptotically vanish in the mean-squared sense. The merits of the general approach are demonstrated in two online learning setups: (i) Robust linear regression using a sparsity-cognizant total least-squares criterion; and (ii) semi-supervised dictionary learning for network-wide link load tracking and imputation with missing entries. Numerical tests on synthetic and real data highlight the potential of the proposed framework for streaming data analytics by demonstrating superior performance over block coordinate descent, and reduced complexity relative to the popular alternating-direction method of multipliers.

cs.LG↗

arXiv2017

Riemannian-geometry-based modeling and clustering of network-wide non-stationary time series: The brain-network case

This paper advocates Riemannian multi-manifold modeling in the context of network-wide non-stationary time-series analysis. Time-series data, collected sequentially over time and across a network, yield features which are viewed as points in or close to a union of multiple submanifolds of a Riemannian manifold, and distinguishing disparate time series amounts to clustering multiple Riemannian submanifolds. To support the claim that exploiting the latent Riemannian geometry behind many statistical features of time series is beneficial to learning from network data, this paper focuses on brain networks and puts forth two feature-generation schemes for network-wide dynamic time series. The first is motivated by Granger-causality arguments and uses an auto-regressive moving average model to map low-rank linear vector subspaces, spanned by column vectors of appropriately defined observability matrices, to points into the Grassmann manifold. The second utilizes (non-linear) dependencies among network nodes by introducing kernel-based partial correlations to generate points in the manifold of positive-definite matrices. Capitilizing on recently developed research on clustering Riemannian submanifolds, an algorithm is provided for distinguishing time series based on their geometrical properties, revealed within Riemannian feature spaces. Extensive numerical tests demonstrate that the proposed framework outperforms classical and state-of-the-art techniques in clustering brain-network states/structures hidden beneath synthetic fMRI time series and brain-activity signals generated from real brain-network structural connectivity matrices.

cs.LG↗

arXiv2015

Large-scale subspace clustering using sketching and validation

The nowadays massive amounts of generated and communicated data present major challenges in their processing. While capable of successfully classifying nonlinearly separable objects in various settings, subspace clustering (SC) methods incur prohibitively high computational complexity when processing large-scale data. Inspired by the random sampling and consensus (RANSAC) approach to robust regression, the present paper introduces a randomized scheme for SC, termed sketching and validation (SkeVa-)SC, tailored for large-scale data. At the heart of SkeVa-SC lies a randomized scheme for approximating the underlying probability density function of the observed data by kernel smoothing arguments. Sparsity in data representations is also exploited to reduce the computational burden of SC, while achieving high clustering accuracy. Performance analysis as well as extensive numerical tests on synthetic and real data corroborate the potential of SkeVa-SC and its competitive performance relative to state-of-the-art scalable SC approaches. Keywords: Subspace clustering, big data, kernel smoothing, randomization, sketching, validation, sparsity.

cs.LG↗

arXiv2015

Sketch and Validate for Big Data Clustering

In response to the need for learning tools tuned to big data analytics, the present paper introduces a framework for efficient clustering of huge sets of (possibly high-dimensional) data. Building on random sampling and consensus (RANSAC) ideas pursued earlier in a different (computer vision) context for robust regression, a suite of novel dimensionality and set-reduction algorithms is developed. The advocated sketch-and-validate (SkeVa) family includes two algorithms that rely on K-means clustering per iteration on reduced number of dimensions and/or feature vectors: The first operates in a batch fashion, while the second sequential one offers computational efficiency and suitability with streaming modes of operation. For clustering even nonlinearly separable vectors, the SkeVa family offers also a member based on user-selected kernel functions. Further trading off performance for reduced complexity, a fourth member of the SkeVa family is based on a divergence criterion for selecting proper minimal subsets of feature variables and vectors, thus bypassing the need for K-means clustering per iteration. Extensive numerical tests on synthetic and real data sets highlight the potential of the proposed algorithms, and demonstrate their competitive performance relative to state-of-the-art random projection alternatives.

stat.ML↗

arXiv2014

Riemannian Multi-Manifold Modeling

This paper advocates a novel framework for segmenting a dataset in a Riemannian manifold $M$ into clusters lying around low-dimensional submanifolds of $M$. Important examples of $M$, for which the proposed clustering algorithm is computationally efficient, are the sphere, the set of positive definite matrices, and the Grassmannian. The clustering problem with these examples of $M$ is already useful for numerous application domains such as action identification in video sequences, dynamic texture clustering, brain fiber segmentation in medical imaging, and clustering of deformed images. The proposed clustering algorithm constructs a data-affinity matrix by thoroughly exploiting the intrinsic geometry and then applies spectral clustering. The intrinsic local geometry is encoded by local sparse coding and more importantly by directional information of local tangent spaces and geodesics. Theoretical guarantees are established for a simplified variant of the algorithm even when the clusters intersect. To avoid complication, these guarantees assume that the underlying submanifolds are geodesic. Extensive validation on synthetic and real data demonstrates the resiliency of the proposed method against deviations from the theoretical model as well as its superior performance over state-of-the-art techniques.

stat.ML↗

arXiv2012

Generalized Thresholding and Online Sparsity-Aware Learning in a Union of Subspaces

This paper studies a sparse signal recovery task in time-varying (time-adaptive) environments. The contribution of the paper to sparsity-aware online learning is threefold; first, a Generalized Thresholding (GT) operator, which relates to both convex and non-convex penalty functions, is introduced. This operator embodies, in a unified way, the majority of well-known thresholding rules which promote sparsity. Second, a non-convexly constrained, sparsity-promoting, online learning scheme, namely the Adaptive Projection-based Generalized Thresholding (APGT), is developed that incorporates the GT operator with a computational complexity that scales linearly to the number of unknowns. Third, the novel family of partially quasi-nonexpansive mappings is introduced as a functional analytic tool for treating the GT operator. By building upon the rich fixed point theory, the previous class of mappings helps us, also, to establish a link between the GT operator and a union of linear subspaces; a non-convex object which lies at the heart of any sparsity promoting technique, batch or online. Based on such a functional analytic framework, a convergence analysis of the APGT is provided. Furthermore, extensive experiments suggest that the APGT exhibits competitive performance when compared to computationally more demanding alternatives, such as the sparsity-promoting Affine Projection Algorithm (APA)- and Recursive Least Squares (RLS)-based techniques.

cs.IT↗