Fourier Sampling in Signal Processing and Numerical Linear Algebra

This thesis focuses on developing efficient algorithmic tools for processing large datasets. In many modern data analysis tasks, the sheer volume of available datasets far outstrips our abilities to process them. This scenario commonly arises in tasks including parameter tuning of machine learning models (e.g., Google Vizier) and training neural networks. These tasks often require solving numerical linear algebraic problems on large matrices, making the classical primitives prohibitively expensive. Hence, there is a crucial need to efficiently compress the available datasets. In other settings, even collecting the input dataset is extremely expensive, making it vital to design optimal data sampling strategies. This is common in applications such as MRI acquisition and spectrum sensing. The fundamental questions above are often dual to each other, and hence can be addressed using the same set of core techniques. Indeed, exploiting structured Fourier sparsity is a recurring source of efficiency in this thesis, leading to both fast numerical linear algebra methods and sample efficient data acquisition schemes.

One of the main results that we present in this thesis is the first Sublinear-time Model-based Sparse FFT algorithm that achieves a nearly optimal sample complexity for recovery of every signal whose Fourier transform is well approximated by a small number of blocks (e.g., such structure is common in spectrum sensing). Our method matches in sublinear time the result of Baraniuk et. al. (2010), which started the field of model-based compressed sensing. Another highlight of this thesis includes the first Dimension-independent Sparse FFT algorithm that, computes the Fourier transform of a sparse signal in sublinear runtime in any dimension. This is the first algorithm that just like the FFT of Cooley and Tukey is dimension independent and avoids the curse of dimensionality inherent to all previously known techniques. Finally, we give a Universal Sampling Scheme for the reconstruction of structured Fourier signals from continuous measurements. Our approach matches the classical results of Slepian, Pollak, and Landau (1960s) on the reconstruction of bandlimited signals via Prolate Spheroidal Wave Functions and extends these results to a wide class of Fourier structure types.

Besides having classical applications in signal processing and data analysis, Fourier techniques have been at the core of many machine learning tasks such as Kernel Matrix Approximation. The second half of this thesis is dedicated to finding compressed and low-rank representations of kernel matrices, which are the primary means of computation with large kernel matrices in machine learning. We build on Fourier techniques and achieve spectral approximation guarantees to the Gaussian kernel using an optimal number of samples, significantly improving upon the classical Random Fourier Features of Rahimi and Recht (2008). Finally, we present a nearly-optimal Oblivious Subspace Embedding for high-degree Polynomial kernels which leads to nearly-optimal embeddings of the high-dimensional Gaussian kernel. This is the first result that does not suffer from an exponential loss in the degree of the polynomial kernel or the dimension of the input point set, providing exponential improvements over the prior works, including the TensorSketch of Pagh (2013) and application of the celebrated Fast Multipole Method of Greengard and Rokhlin (1986) to the kernel approximation problem.

Timing is Everything

Karen Adam

In this thesis, timing is everything. In the first part, we mean this literally, as we tackle systems that encode information using timing alone. In the second part, we adopt the standard, metaphoric interpretation of this saying and show the importance of choosing the right time to sample a system, when efficiency is a key consideration.Time encoding machines, or, alternately, integrate-and-fire neurons, encode their inputs using spikes, the timings of which depend on the input and therefore hold information about it. These devices can be made more power efficient than their clocked counterparts and have thus been studied in the fields of signal processing, event-based vision, computational neuroscience and machine learning. However, their timing-based spiking output has so far often been considered a nuisance that one must make do with, rather than a potential advantage. In this thesis, we show that this timing-based output equips spiking devices with capabilities that are out of reach for classical encoding and processing systems.We first discover the benefits of time encoding on multi-channel encoding and recovery of a signal: with time encoding, clock alignment is easy to solve, although it poses problems in the classical sampling scenario. Then, we study the time encoding of low-dimensional signals and see that the asynchrony of spikes allows for a lower sample complexity in comparison with synchronous sampling. Thanks to this same asynchrony, time encoding of video results in an entanglement between spatial sampling density and temporal resolution---a relationship which is not present in frame-based video. Finally, we show that the all-or-none nature of the spikes allows training spiking neural networks in a layer-by-layer fashion---a feat that is impossible with clocked, artificial neural networks, due to the credit assignment problem.The second part of this thesis shows that choosing the right timing of samples can be crucial to ensure efficiency when performing nanoscale magnetic sensing. We are given a stochastic process, where each sample at time t follows a Bernoulli distribution, and which is characterized by oscillation frequencies that we are interested in recovering. We search for an optimal approach to sample this process, such that the variance of the frequencies' estimates is minimized, given constraints on the measurement time. The models we assume stem from the field of nanoscale magnetic sensing, where the number of parameters to be estimated varies with the number of spins one is trying to sense. We present an adaptive approach to choosing samples in both the single-spin and two-spin cases and compare the adaptive algorithm's performance to classical approaches to sampling.In both parts of the thesis, we move away from classical amplitude sampling and consider cases where timing takes the forefront and amplitude information is merely binary, to shows that timing can carry information and that it can control the amount of information gain.

EPFL2022

Adaptive wake and sleep detection for wearable systems

Walter Karlen

Sleep problems and disorders have a serious impact on human health and wellbeing. The rising costs for treating sleep-related chronic diseases in industrialized countries demands efficient prevention. Low-cost, wearable sleep / wake detection systems which give feedback on the wearer's "sleep performance" are a promising approach to reduce the risk of developing serious sleep disorders and fatigue. Not all bio-medical signals that are useful for sleep / wake discrimination can be easily recorded with wearable systems. Sensors often need to be placed in an obtrusive location on the body or cannot be efficiently embedded into a wearable frame. Furthermore, wearable systems have limited computational and energetic resources, which restrict the choice of sensors and algorithms for online processing and classification. Since wearable systems are used outside the laboratory, the recorded signals tend to be corrupted with additional noise that influences the precision of classification algorithms. In this thesis we present the research on a wearable sleep / wake classifier system that relies on cardiorespiratory (ECG and respiratory effort) and activity recordings and that works autonomously with minimal user interaction. This research included the selection of optimal signals and sensors, the development of a custom-tailored hardware demonstrator with embedded classification algorithms, and the realization of experiments in real-world environments for the customization and validation of the system. The processing and classification of the signals were based on Fourier transformations and artificial neural networks that are efficiently implementable into digital signal controllers. Literature analysis and empiric measurements revealed that cardiorespiratory signals are more promising for a wearable sleep / wake classification than clinically used signals such as brain potentials. The experiments conducted during this thesis showed that inter-subject differences within the recorded physiological signals make it difficult to design a sleep / wake classification model that can generalize to a group of subjects. This problem was addressed in two ways: First by adding features from another signal to the classifier, that is, measuring the behavioral quiescence during sleep using accelerometers. Conducted research on different feature extraction methods from accelerometer data showed that this data generalizes well for distinct subjects in the study group. In addition, research on user-adaptation methods was conducted. Behavioral sleep and wake measures, notably the measurement of reactivity and activity, were developed to build up a priori knowledge that was used to adapt the classification algorithm automatically to new situations. This thesis demonstrates the design and development of a low-cost, wearable hardware and embedded software for on-line sleep / wake discrimination. The proposed automatic user-adaptive classifier is advantageous compared to previously suggested classification methods that generalize over multiple subjects, because it can take changes in the wearer's physiology and sleep / wake behavior into account without adjustment from a human expert. The results of this thesis contribute to the development of smart, wearable, bio-physiological monitoring systems which require a high degree of autonomy and have only low computational resources available. We believe that the proposed sleep / wake classification system is a first promising step toward a context-aware system for sleep management, sleep disorder prevention, and reduction of fatigue.

EPFL2009

Signal structure: from manifolds to molecules and structured sparsity

Sofia Karygianni

Effective representation methods and proper signal priors are crucial in most signal processing applications. In this thesis we focus on different structured models and we design appropriate schemes that allow the discovery of low dimensional latent structures that characterise and identify the signals. Motivated by the highly non-linear structure of most datasets, we firstly investigate the geometry of manifolds. Manifolds are low dimensional, non-linear structures that are naturally employed to describe sets of strongly related signals such as the images of a 3-D object captured from different viewpoints. However, the use of manifolds in applications is not straightforward due to their usually non-analytic and non-linear form. We propose here a way to `disassemble' a manifold into simpler components by approximating it with affine subspaces. Our objective is to discover a set of low dimensional affine subspaces that can represent manifold data accurately while preserving the manifold's structure. To this end, we employ a greedy technique that iteratively merges manifold samples into groups based on the difference of local tangents. We use our algorithm to approximate synthetic and real manifolds and to demonstrate that it is competitive to state-of-the-art techniques. Then, we consider structured sparse representations of signals and we propose a new sparsity model, where signals are essentially composed of a small number of structured {\it molecules }. We define the molecules to be linear combinations of a small number of atoms in a redundant dictionary. Our multi-level model takes into account the energy distribution of the significant signal components in addition to their support. It permits to define typical visual patterns and recognise them in prototypical or deformed form. We define a new structural difference measure between molecules and their deformed versions, which is based on their sparse codes and we create an algorithm for decomposing signals into molecules that can account for different deviations in the internal molecule structure. Our experiments verify the benefits of the new image model in various restoration tasks and they confirm that the development of proper models that extend the mere notion of sparsity can be very useful for various inverse problems in imaging. Finally, we investigate the problem of learning molecule representations directly in the sparse code domain. We constrain sparse codes to be linear combinations of a few, possibly deformed, molecules and we design an algorithm that can learn the structure from the codes without transforming them back into the signal domain. To this end, we take advantage of our structural difference which is based on the sparse codes and we devise a scheme for representing the codes with molecules and learn the molecules at the same time. To illustrate the effectiveness of our proposed algorithm we apply it to various synthetic and real datasets and we compare the results with traditional sparse coding and dictionary learning techniques. From the experiments, we verify the superior performance of our scheme in interpreting and recognising correctly the underlying structure. In short, in this thesis we are interested in low-dimensional, structured models. Among the various choices, we focus on manifolds and sparse representations and we propose schemes that enhance their structural properties and highlight their effectiveness in signal representations.

EPFL2015