arXiv subjects

Helmut Bölcskei

Publications and source records attributed to Helmut Bölcskei.

At least 37 records · Page 2Linked to original sources

arXiv2024

Three Quantization Regimes for ReLU Networks

We establish the fundamental limits in the approximation of Lipschitz functions by deep ReLU neural networks with finite-precision weights. Specifically, three regimes, namely under-, over-, and proper quantization, in terms of minimax approximation error behavior as a function of network weight precision, are identified. This is accomplished by deriving nonasymptotic tight lower and upper bounds on the minimax approximation error. Notably, in the proper-quantization regime, neural networks exhibit memory-optimality in the approximation of Lipschitz functions. Deep networks have an inherent advantage over shallow networks in achieving memory-optimality. We also develop the notion of depth-precision tradeoff, showing that networks with high-precision weights can be converted into functionally equivalent deeper networks with low-precision weights, while preserving memory-optimality. This idea is reminiscent of sigma-delta analog-to-digital conversion, where oversampling rate is traded for resolution in the quantization of signal samples. We improve upon the best-known ReLU network approximation results for Lipschitz functions and describe a refinement of the bit extraction technique which could be of independent general interest.

stat.ML↗

arXiv2024

Cellular automata, many-valued logic, and deep neural networks

We develop a theory characterizing the fundamental capability of deep neural networks to learn, from evolution traces, the logical rules governing the behavior of cellular automata (CA). This is accomplished by first establishing a novel connection between CA and Lukasiewicz propositional logic. While binary CA have been known for decades to essentially perform operations in Boolean logic, no such relationship exists for general CA. We demonstrate that many-valued (MV) logic, specifically Lukasiewicz propositional logic, constitutes a suitable language for characterizing general CA as logical machines. This is done by interpolating CA transition functions to continuous piecewise linear functions, which, by virtue of the McNaughton theorem, yield formulae in MV logic characterizing the CA. Recognizing that deep rectified linear unit (ReLU) networks realize continuous piecewise linear functions, it follows that these formulae are naturally extracted from CA evolution traces by deep ReLU networks. A corresponding algorithm together with a software implementation is provided. Finally, we show that the dynamical behavior of CA can be realized by recurrent neural networks.

cs.AI↗

arXiv2023

Lossy Compression of General Random Variables

This paper is concerned with the lossy compression of general random variables, specifically with rate-distortion theory and quantization of random variables taking values in general measurable spaces such as, e.g., manifolds and fractal sets. Manifold structures are prevalent in data science, e.g., in compressed sensing, machine learning, image processing, and handwritten digit recognition. Fractal sets find application in image compression and in the modeling of Ethernet traffic. Our main contributions are bounds on the rate-distortion function and the quantization error. These bounds are very general and essentially only require the existence of reference measures satisfying certain regularity conditions in terms of small ball probabilities. To illustrate the wide applicability of our results, we particularize them to random variables taking values in i) manifolds, namely, hyperspheres and Grassmannians, and ii) self-similar sets characterized by iterated function systems satisfying the weak separation property.

math.PR↗

arXiv2022

Metric entropy of causal, discrete-time LTI systems

In [1] it is shown that recurrent neural networks (RNNs) can learn - in a metric entropy optimal manner - discrete time, linear time-invariant (LTI) systems. This is effected by comparing the number of bits needed to encode the approximating RNN to the metric entropy of the class of LTI systems under consideration [2, 3]. The purpose of this note is to provide an elementary self-contained proof of the metric entropy results in [2, 3], in the process of which minor mathematical issues appearing in [2, 3] are cleaned up. These corrections also lead to the correction of a constant in a result in [1] (see Remark 2.5).

math.DS↗

arXiv2022

High-Dimensional Distribution Generation Through Deep Neural Networks

We show that every $d$-dimensional probability distribution of bounded support can be generated through deep ReLU networks out of a $1$-dimensional uniform input distribution. What is more, this is possible without incurring a cost - in terms of approximation error measured in Wasserstein-distance - relative to generating the $d$-dimensional target distribution from $d$ independent random variables. This is enabled by a vast generalization of the space-filling approach discovered in (Bailey & Telgarsky, 2018). The construction we propose elicits the importance of network depth in driving the Wasserstein distance between the target distribution and its neural network approximation to zero. Finally, we find that, for histogram target distributions, the number of bits needed to encode the corresponding generative network equals the fundamental limit for encoding probability distributions as dictated by quantization theory.

cs.LG↗

arXiv2021

Metric Entropy Limits on Recurrent Neural Network Learning of Linear Dynamical Systems

One of the most influential results in neural network theory is the universal approximation theorem [1, 2, 3] which states that continuous functions can be approximated to within arbitrary accuracy by single-hidden-layer feedforward neural networks. The purpose of this paper is to establish a result in this spirit for the approximation of general discrete-time linear dynamical systems - including time-varying systems - by recurrent neural networks (RNNs). For the subclass of linear time-invariant (LTI) systems, we devise a quantitative version of this statement. Specifically, measuring the complexity of the considered class of LTI systems through metric entropy according to [4], we show that RNNs can optimally learn - or identify in system-theory parlance - stable LTI systems. For LTI systems whose input-output relation is characterized through a difference equation, this means that RNNs can learn the difference equation from input-output traces in a metric-entropy optimal manner.

cs.LG↗

arXiv2021

Constructive Universal High-Dimensional Distribution Generation through Deep ReLU Networks

We present an explicit deep neural network construction that transforms uniformly distributed one-dimensional noise into an arbitrarily close approximation of any two-dimensional Lipschitz-continuous target distribution. The key ingredient of our design is a generalization of the "space-filling" property of sawtooth functions discovered in (Bailey & Telgarsky, 2018). We elicit the importance of depth - in our neural network construction - in driving the Wasserstein distance between the target distribution and the approximation realized by the network to zero. An extension to output distributions of arbitrary dimension is outlined. Finally, we show that the proposed construction does not incur a cost - in terms of error measured in Wasserstein-distance - relative to generating $d$-dimensional target distributions from $d$ independent random variables.

cs.LG↗

arXiv2021

Deep Neural Network Approximation Theory

This paper develops fundamental limits of deep neural network learning by characterizing what is possible if no constraints are imposed on the learning algorithm and on the amount of training data. Concretely, we consider Kolmogorov-optimal approximation through deep neural networks with the guiding theme being a relation between the complexity of the function (class) to be approximated and the complexity of the approximating network in terms of connectivity and memory requirements for storing the network topology and the associated quantized weights. The theory we develop establishes that deep networks are Kolmogorov-optimal approximants for markedly different function classes, such as unit balls in Besov spaces and modulation spaces. In addition, deep networks provide exponential approximation accuracy - i.e., the approximation error decays exponentially in the number of nonzero weights in the network - of the multiplication operation, polynomials, sinusoidal functions, and certain smooth functions. Moreover, this holds true even for one-dimensional oscillatory textures and the Weierstrass function - a fractal function, neither of which has previously known methods achieving exponential approximation accuracy. We also show that in the approximation of sufficiently smooth functions finite-width deep networks require strictly smaller connectivity than finite-depth wide networks.

cs.LG↗

arXiv2020

Affine symmetries and neural network identifiability

We address the following question of neural network identifiability: Suppose we are given a function $f:\mathbb{R}^m\to\mathbb{R}^n$ and a nonlinearity $ρ$. Can we specify the architecture, weights, and biases of all feed-forward neural networks with respect to $ρ$ giving rise to $f$? Existing literature on the subject suggests that the answer should be yes, provided we are only concerned with finding networks that satisfy certain "genericity conditions". Moreover, the identified networks are mutually related by symmetries of the nonlinearity. For instance, the $\tanh$ function is odd, and so flipping the signs of the incoming and outgoing weights of a neuron does not change the output map of the network. The results known hitherto, however, apply either to single-layer networks, or to networks satisfying specific structural assumptions (such as full connectivity), as well as to specific nonlinearities. In an effort to answer the identifiability question in greater generality, we consider arbitrary nonlinearities with potentially complicated affine symmetries, and we show that the symmetries can be used to find a rich set of networks giving rise to the same function $f$. The set obtained in this manner is, in fact, exhaustive (i.e., it contains all networks giving rise to $f$) unless there exists a network $\mathcal{A}$ "with no internal symmetries" giving rise to the identically zero function. This result can thus be interpreted as an analog of the rank-nullity theorem for linear operators. We furthermore exhibit a class of "$\tanh$-type" nonlinearities (including the tanh function itself) for which such a network $\mathcal{A}$ does not exist, thereby solving the identifiability question for these nonlinearities in full generality. Finally, we show that this class contains nonlinearities with arbitrarily complicated symmetries.

cs.IT↗

arXiv2020

Neural network identifiability for a family of sigmoidal nonlinearities

This paper addresses the following question of neural network identifiability: Does the input-output map realized by a feed-forward neural network with respect to a given nonlinearity uniquely specify the network architecture, weights, and biases? Existing literature on the subject Sussman 1992, Albertini, Sontag et al. 1993, Fefferman 1994 suggests that the answer should be yes, up to certain symmetries induced by the nonlinearity, and provided the networks under consideration satisfy certain "genericity conditions". The results in Sussman 1992 and Albertini, Sontag et al. 1993 apply to networks with a single hidden layer and in Fefferman 1994 the networks need to be fully connected. In an effort to answer the identifiability question in greater generality, we derive necessary genericity conditions for the identifiability of neural networks of arbitrary depth and connectivity with an arbitrary nonlinearity. Moreover, we construct a family of nonlinearities for which these genericity conditions are minimal, i.e., both necessary and sufficient. This family is large enough to approximate many commonly encountered nonlinearities to within arbitrary precision in the uniform norm.

math.CO↗

arXiv2020

Canonical Conditions for K/2 Degrees of Freedom

We present a necessary and sufficient condition for $1/2$ degree of freedom for each user in constant $K$-user single-antenna interference channels. This condition applies to all channel topologies, i.e., to fully-connected channels as well as channels that have individual links absent, reflected by corresponding zeros in the channel matrix. Moreover, it captures the essence of interference alignment by virtue of being expressed in terms of a generic injectivity condition that guarantees separability of signal and interference. Finally, we provide codebook constructions achieving $1/2$ degree of freedom for each user for all channel matrices satisfying our condition.

cs.IT↗

arXiv2020

Uncertainty relations and sparse signal recovery

This chapter provides a principled introduction to uncertainty relations underlying sparse signal recovery. We start with the seminal work by Donoho and Stark, 1989, which defines uncertainty relations as upper bounds on the operator norm of the band-limitation operator followed by the time-limitation operator, generalize this theory to arbitrary pairs of operators, and then develop -- out of this generalization -- the coherence-based uncertainty relations due to Elad and Bruckstein, 2002, as well as uncertainty relations in terms of concentration of $1$-norm or $2$-norm. The theory is completed with the recently discovered set-theoretic uncertainty relations which lead to best possible recovery thresholds in terms of a general measure of parsimony, namely Minkowski dimension. We also elaborate on the remarkable connection between uncertainty relations and the "large sieve", a family of inequalities developed in analytic number theory. It is finally shown how uncertainty relations allow to establish fundamental limits of practical signal recovery problems such as inpainting, declipping, super-resolution, and denoising of signals corrupted by impulse noise or narrowband interference. Detailed proofs are provided throughout the chapter.

cs.IT↗

arXiv2018

Noisy subspace clustering via matching pursuits

Sparsity-based subspace clustering algorithms have attracted significant attention thanks to their excellent performance in practical applications. A prominent example is the sparse subspace clustering (SSC) algorithm by Elhamifar and Vidal, which performs spectral clustering based on an adjacency matrix obtained by sparsely representing each data point in terms of all the other data points via the Lasso. When the number of data points is large or the dimension of the ambient space is high, the computational complexity of SSC quickly becomes prohibitive. Dyer et al. observed that SSC-OMP obtained by replacing the Lasso by the greedy orthogonal matching pursuit (OMP) algorithm results in significantly lower computational complexity, while often yielding comparable performance. The central goal of this paper is an analytical performance characterization of SSC-OMP for noisy data. Moreover, we introduce and analyze the SSC-MP algorithm, which employs matching pursuit (MP) in lieu of OMP. Both SSC-OMP and SSC-MP are proven to succeed even when the subspaces intersect and when the data points are contaminated by severe noise. The clustering conditions we obtain for SSC-OMP and SSC-MP are similar to those for SSC and for the thresholding-based subspace clustering (TSC) algorithm due to Heckel and Bölcskei. Analytical results in combination with numerical results indicate that both SSC-OMP and SSC-MP with a data-dependent stopping criterion automatically detect the dimensions of the subspaces underlying the data. Moreover, experiments on synthetic and on real data show that SSC-MP compares very favorably to SSC, SSC-OMP, TSC, and the nearest subspace neighbor algorithm, both in terms of clustering performance and running time. In addition, we find that, in contrast to SSC-OMP, the performance of SSC-MP is very robust with respect to the choice of parameters in the stopping criteria.

cs.LG↗

arXiv2018

The universal approximation power of finite-width deep ReLU networks

We show that finite-width deep ReLU neural networks yield rate-distortion optimal approximation (Bölcskei et al., 2018) of polynomials, windowed sinusoidal functions, one-dimensional oscillatory textures, and the Weierstrass function, a fractal function which is continuous but nowhere differentiable. Together with their recently established universal approximation property of affine function systems (Bölcskei et al., 2018), this shows that deep neural networks approximate vastly different signal structures generated by the affine group, the Weyl-Heisenberg group, or through warping, and even certain fractals, all with approximation error decaying exponentially in the number of neurons. We also prove that in the approximation of sufficiently smooth functions finite-width deep networks require strictly smaller connectivity than finite-depth wide networks.

cs.LG↗

arXiv2018

Optimal Approximation with Sparsely Connected Deep Neural Networks

We derive fundamental lower bounds on the connectivity and the memory requirements of deep neural networks guaranteeing uniform approximation rates for arbitrary function classes in $L^2(\mathbb R^d)$. In other words, we establish a connection between the complexity of a function class and the complexity of deep neural networks approximating functions from this class to within a prescribed accuracy. Additionally, we prove that our lower bounds are achievable for a broad family of function classes. Specifically, all function classes that are optimally approximated by a general class of representation systems---so-called \emph{affine systems}---can be approximated by deep neural networks with minimal connectivity and memory requirements. Affine systems encompass a wealth of representation systems from applied harmonic analysis such as wavelets, ridgelets, curvelets, shearlets, $α$-shearlets, and more generally $α$-molecules. Our central result elucidates a remarkable universality property of neural networks and shows that they achieve the optimum approximation properties of all affine systems combined. As a specific example, we consider the class of $α^{-1}$-cartoon-like functions, which is approximated optimally by $α$-shearlets. We also explain how our results can be extended to the case of functions on low-dimensional immersed manifolds. Finally, we present numerical experiments demonstrating that the standard stochastic gradient descent algorithm generates deep neural networks providing close-to-optimal approximation rates. Moreover, these results indicate that stochastic gradient descent can actually learn approximations that are sparse in the representation systems optimally sparsifying the function class the network is trained on.

cs.LG↗

arXiv2018

Necessary Conditions for K/2 Degrees of Freedom

Stotz et al., 2016, reported a sufficient (injectivity) condition for each user in a K-user single-antenna constant interference channel to achieve 1/2 degree of freedom. The present paper proves that this condition is necessary as well and hence provides an equivalence characterization of interference channel matrices allowing full degrees of freedom.

cs.IT↗

arXiv2018

Rate-Distortion Theory for General Sets and Measures

This paper is concerned with a rate-distortion theory for sequences of i.i.d. random variables with general distribution supported on general sets including manifolds and fractal sets. Manifold structures are prevalent in data science, e.g., in compressed sensing, machine learning, image processing, and handwritten digit recognition. Fractal sets find application in image compression and in modeling of Ethernet traffic. We derive a lower bound on the (single-letter) rate-distortion function that applies to random variables X of general distribution and for continuous X reduces to the classical Shannon lower bound. Moreover, our lower bound is explicit up to a parameter obtained by solving a convex optimization problem in a nonnegative real variable. The only requirement for the bound to apply is the existence of a sigma-finite reference measure for X satisfying a certain subregularity condition. This condition is very general and prevents the reference measure from being highly concentrated on balls of small radii. To illustrate the wide applicability of our result, we evaluate the lower bound for a random variable distributed uniformly on a manifold, namely, the unit circle, and a random variable distributed uniformly on a self-similar set, namely, the middle third Cantor set.

cs.IT↗

arXiv2018

Topology Reduction in Deep Convolutional Feature Extraction Networks

Deep convolutional neural networks (CNNs) used in practice employ potentially hundreds of layers and $10$,$000$s of nodes. Such network sizes entail significant computational complexity due to the large number of convolutions that need to be carried out; in addition, a large number of parameters needs to be learned and stored. Very deep and wide CNNs may therefore not be well suited to applications operating under severe resource constraints as is the case, e.g., in low-power embedded and mobile platforms. This paper aims at understanding the impact of CNN topology, specifically depth and width, on the network's feature extraction capabilities. We address this question for the class of scattering networks that employ either Weyl-Heisenberg filters or wavelets, the modulus non-linearity, and no pooling. The exponential feature map energy decay results in Wiatowski et al., 2017, are generalized to $\mathcal{O}(a^{-N})$, where an arbitrary decay factor $a>1$ can be realized through suitable choice of the Weyl-Heisenberg prototype function or the mother wavelet. We then show how networks of fixed (possibly small) depth $N$ can be designed to guarantee that $((1-\varepsilon)\cdot 100)\%$ of the input signal's energy are contained in the feature vector. Based on the notion of operationally significant nodes, we characterize, partly rigorously and partly heuristically, the topology-reducing effects of (effectively) band-limited input signals, band-limited filters, and feature map symmetries. Finally, for networks based on Weyl-Heisenberg filters, we determine the prototype function bandwidth that minimizes---for fixed network depth $N$---the average number of operationally significant nodes per layer.

stat.ML↗