Most multiobject tracking methods rely on a statistical model that involves unknown parameters. Here, we propose a Bayesian method for class-aided online learning of model parameters within extended multiobject tracking. We address the case where the extended objects belong to unknown object classes defined by unknown values of the model parameters. The proposed method learns the number of object classes, the class parameters, and the objects’ class affiliations simultaneously with the tracking process, and the learned class and parameter information is leveraged for improved tracking. This is enabled by a parameter-dependent state-space model for extended multiobject tracking that incorporates a Dirichlet process prior, and by a related Gibbs sampler for online learning. Our simulation results demonstrate substantial gains in tracking performance due to class-aided online parameter learning.

The constant center frequency to bandwidth ratio (Q-factor) of wavelet transforms provides a very natural representation for audio data. However, invertible wavelet transforms have either required non-uniform decimation—leading to irregular data structures that are cumbersome to work with—or require excessively high oversampling with unacceptable computational overhead. Here, we present a novel decimation strategy for wavelet transforms that leads to stable representations with oversampling rates close to one and uniform decimation. Specifically, we show that finite implementations of the resulting representation are energy-preserving in the sense of frame theory. The obtained wavelet coefficients can be stored in a time-frequency matrix with a natural interpretation of columns as time frames and rows as frequency channels. This matrix structure immediately grants access to a large number of algorithms that are successfully used in time-frequency audio processing, but could not previously be used jointly with wavelet transforms. We demonstrate the application of our method in processing based on nonnegative matrix factorization, in onset detection, and in phaseless reconstruction.

Mutual Information (MI) has been widely used as a loss regularizer for training neural networks. This has been particularly effective when learn disentangled or compressed representations of high dimensional data. However, differential entropy (DE), another fundamental measure of information, has not found widespread use in neural network training. Although DE offers a potentially wider range of applications than MI, off-the-shelf DE estimators are either non differentiable, computationally intractable or fail to adapt to changes in the underlying distribution. These drawbacks prevent them from being used as regularizers in neural networks training. To address shortcomings in previously proposed estimators for DE, here we introduce KNIFE, a fully parameterized, differentiable kernel-based estimator of DE. The flexibility of our approach also allows us to construct KNIFE-based estimators for conditional (on either discrete or continuous variables) DE, as well as MI. We empirically validate our method on high-dimensional synthetic data and further apply it to guide the training of neural networks for real-world tasks. Our experiments on a large variety of tasks, including visual domain adaptation, textual fair classification, and textual fine-tuning demonstrate the effectiveness of KNIFE-based estimation. Code can be found at https://github.com/g-pichler/knife.

Fusing probabilistic information is a fundamental task in signal and data processing with relevance to many fields of technology and science. In this work, we investigate the fusion of multiple probability density functions (pdfs) of a continuous random variable or vector. Although the case of continuous random variables and the problem of pdf fusion frequently arise in multisensor signal processing, statistical inference, and machine learning, a universally accepted method for pdf fusion does not exist. The diversity of approaches, perspectives, and solutions related to pdf fusion motivates a unified presentation of the theory and methodology of the field. We discuss three different approaches to fusing pdfs. In the axiomatic approach, the fusion rule is defined indirectly by a set of properties (axioms). In the optimization approach, it is the result of minimizing an objective function that involves an information-theoretic divergence or a distance measure. In the supra-Bayesian approach, the fusion center interprets the pdfs to be fused as random observations. Our work is partly a survey, reviewing in a structured and coherent fashion many of the concepts and methods that have been developed in the literature. In addition, we present new results for each of the three approaches. Our original contributions include new fusion rules, axioms, and axiomatic and optimization-based characterizations; a new formulation of supra-Bayesian fusion in terms of finite-dimensional parametrizations; and a study of supra-Bayesian fusion of posterior pdfs for linear Gaussian models.

We introduce Minimal Achievable Sufficient Statistic (MASS) Learning, a machine learning training objective for which the minima are minimal sufficient statistics with respect to a class of functions being optimized over (e.g., deep networks). In deriving MASS Learning, we also introduce Conserved Differential Information (CDI), an information-theoretic quantity that {—} unlike standard mutual information {—} can be usefully applied to deterministically-dependent continuous random variables like the input and output of a deep network. In a series of experiments, we show that deep networks trained with MASS Learning achieve competitive performance on supervised learning, regularization, and uncertainty quantification benchmarks.

We establish the fundamental limits of lossless analog compression by considering the recovery of arbitrary random vectors x ∈ Rmfrom the noiseless linear measurements y = Ax with measurement matrix A ∈ Rn×m. Our theory is inspired by the groundbreaking work of Wu and Verdú (2010) on almost lossless analog compression but applies to the nonasymptotic, i.e., fixed-m case, and considers zero error probability. Specifically, our achievability result states that, for Lebesgue-almost all A, the random vector x can be recovered with zero error probability provided that n > K(x), where K(x) is given by the infimum of the lower modified Minkowski dimension over all support sets U of x (i.e., sets U ⊆ Rmwith P[x ∈ U] = 1). We then particularize this achievability result to the class of s-rectifiable random vectors as introduced in Koliander et al. (2016); these are random vectors of absolutely continuous distribution-with respect to the s-dimensional Hausdorff measure-supported on countable unions of s-dimensional C1-submanifolds of Rm. Countable unions of C1-submanifolds include essentially all signal models used in the compressed sensing literature such as the standard union of subspaces model underlying much of compressed sensing theory and spectrum-blind sampling, smooth submanifolds, block-sparsity, and low-rank matrices as considered in the matrix completion problem. Specifically, we prove that, for Lebesgue-almost all A, s-rectifiable random vectors x can be recovered with zero error probability from n > s linear measurements. This threshold is, however, found not to be tight as exemplified by the construction of an s-rectifiable random vector that can be recovered with zero error probability from n <; s linear measurements. Motivated by this observation, we introduce the new class of s-analytic random vectors, which admit a strong converse in the sense of n ≥ s being necessary for recovery with probability of error smaller than one. The central conceptual tools in the development of our theory are geometric measure theory and the theory of real analytic functions.

We prove rigorously a source coding theorem that can probably be considered folklore, a generalization to arbitrary alphabets of a problem motivated by the Information Bottleneck method. For general random variables (Y, X), we show essentially that for some n ∈ \mathbbN, a function f with rate limit log|f| ≤ nR and I(Yn;f(Xn)) ≥ nS exists if and only if there is a random variable U such that the Markov chain Y°-X°-U holds, I(U;X) ≤ R and I(U;Y) ≥ S. The proof relies on the well established discrete case and showcases a technique for lifting discrete coding theorems to arbitrary alphabets.

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$\mu_{X}$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$\mu$, for$X$(i.e., a measure$\mu$with$\mu x\ll\mu$and such that the generalized entropy$h_{\mu}(X)$is finite) satisfying a certain subregularity condition. This condition is very general and prevents the reference measure$\mu$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.

We study the compression of data in the case where the useful information is contained in a set rather than a vector, i.e., the ordering of the data points is irrelevant and the number of data points is unknown. Our analysis is based on rate-distortion theory and the theory of finite point processes. We introduce fundamental information-theoretic concepts and quantities for point processes and present general lower and upper bounds on the rate-distortion function. To enable a comparison with the vector setting, we concretize our bounds for point processes of fixed cardinality. In particular, we analyze a fixed number of unordered Gaussian data points and show that we can significantly reduce the required rates compared to the best possible compression strategy for Gaussian vectors. As an example of point processes with variable cardinality, we study the best possible compression of Poisson point processes. For the specific case of a Poisson point process with uniform intensity on the unit square, our lower and upper bounds are separated by only a small gap and thus provide a good characterization of the rate-distortion function.

We propose a method for detecting and estimating multiple objects from multiple noisy images with partly overlapping observation areas. The goal is to detect the objects that are “locally” present in the individual observation areas and to estimate their states. Our method is based on a new closed-form expression of the marginal posterior probability hypothesis density (PHD) and admits a distributed implementation. Simulation results demonstrate performance gains over correlation-based and PHD-based methods that do not take advantage of the overlapping observation areas.

We establish the fundamental limits of lossless linear analog compression by considering the recovery of random vectors x ∈ ℝmfrom the noiseless linear measurements y = Ax with measurement matrix A ∈ ℝn×m. Specifically, for a random vector x ∈ ℝmof arbitrary distribution we show that x can be recovered with zero error probability from n > inf dimMB(U) linear measurements, where dimMB(·) denotes the lower modified Minkowski dimension and the infimum is over all sets U ⊆ ℝmwith P[x ∈ U] = 1. This achievability statement holds for Lebesgue almost all measurement matrices A. We then show that s-rectifiable random vectors-a stochastic generalization of s-sparse vectors-can be recovered with zero error probability from n > s linear measurements. From classical compressed sensing theory we would expect n ≥ s to be necessary for successful recovery of x. Surprisingly, certain classes of s-rectifiable random vectors can be recovered from fewer than s measurements. Imposing an additional regularity condition on the distribution of s-rectifiable random vectors x, we do get the expected converse result of s measurements being necessary. The resulting class of random vectors appears to be new and will be referred to as s-analytic random vectors.

Entropy and differential entropy are important quantities in information theory. A tractable extension to singular random variables-which are neither discrete nor continuous- has not been available so far. Here, we present such an extension for the practically relevant class of integer-dimensional singular random variables. The proposed entropy definition contains the entropy of discrete random variables and the differential entropy of continuous random variables as special cases. We show that it transforms in a natural manner under Lipschitz functions, and that it is invariant under unitary transformations. We define joint entropy and conditional entropy for integer-dimensional singular random variables, and we show that the proposed entropy conveys useful expressions of the mutual information. As first applications of our entropy definition, we present a result on the minimal expected codeword length of quantized integer-dimensional singular sources and a Shannon lower bound for integer-dimensional singular sources.

Entropy and differential entropy are important quantities in information theory. A tractable extension to singular random variables (which are neither discrete nor continuous) has not been available so far. Here, we propose such an extension for the practically relevant class of singular probability measures that are supported on a lower-dimensional subset of Euclidean space. We show that our entropy transforms in a natural manner under Lipschitz functions and that it conveys useful expressions of the mutual information. Potential applications of the proposed entropy definition include capacity calculations for the vector interference channel, compressed sensing in a probabilistic setting, and capacity bounds for block-fading channel models.

We analyze the capacity of a continuous-time, time-selective, Rayleigh block-fading channel in the high signal-to-noise ratio (SNR) regime. The fading process is assumed stationary within each block and to change independently from block to block; furthermore, its realizations are not known a priori to the transmitter and the receiver (noncoherent setting). A common approach to analyzing the capacity of this channel is to assume that the receiver performs matched filtering followed by sampling at symbol rate (symbol matched filtering). This yields a discrete-time channel in which each transmitted symbol corresponds to one output sample. Liang & Veeravalli (2004) showed that the capacity of this discrete-time channel grows logarithmically with the SNR, with a capacity pre-log equal to 1-Q/N. Here, N is the number of symbols transmitted within one fading block, and Q is the rank of the covariance matrix of the discrete-time channel gains within each fading block. In this paper, we show that symbol matched filtering is not a capacity-achieving strategy for the underlying continuous-time channel. Specifically, we analyze the capacity pre-log of the discrete-time channel obtained by oversampling the continuous-time channel output, i.e., by sampling it faster than at symbol rate. We prove that by oversampling by a factor two one gets a capacity pre-log that is at least as large as 1-1/N. Since the capacity pre-log corresponding to symbol-rate sampling is 1-Q/N, our result implies indeed that symbol matched filtering is not capacity achieving at high SNR.

We study the high-signal-to-noise-ratio capacity of generic multiple-input multiple-output (MIMO) Rayleigh block-fading channels in the noncoherent setting where neither transmitter nor receiver has a priori channel state information but both are aware of the channel statistics. In contrast to the well-established constant block-fading model, we allow the fading to vary within each block with a temporal correlation that is generic (in the sense used in the interference-alignment literature). We show that the number of degrees of freedom of a generic MIMO Rayleigh block-fading channel with T transmit antennas and block length N is given by T(1 - 1/N) provided that T <; N and the number of receive antennas is at least T(N - 1)/(N - T). A comparison with the constant block-fading channel (where the fading is constant within each block) shows that, for large block lengths, generic correlation increases the number of degrees of freedom by a factor of up to four.

We study the high-SNR capacity of MIMO Rayleigh block-fading channels in the noncoherent setting where neither transmitter nor receiver has a priori channel state information. We show that when the number of receive antennas is sufficiently large and the temporal correlation within each block is “generic” (in the sense used in the interference-alignment literature), the capacity pre-log is given by T(1 - 1/N) for T <; N, where T denotes the number of transmit antennas and N denotes the block length. A comparison with the widely used constant block-fading channel (where the fading is constant within each block) shows that for a large block length, generic correlation increases the capacity pre-log by a factor of about four.