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Bayesian nonparametric hierarchical priors provide flexible models for sharing of information within and across groups. We focus on latent feature allocation models, where the data structures correspond to multisets or unbounded sparse matrices. The fundamental development in this regard is the Hierarchical Indian Buffet process (HIBP), devised by Thibaux and Jordan (2007). However, little is known in terms of explicit tractable descriptions of the joint, marginal, posterior and predictive distributions of the HIBP. We provide explicit novel descriptions of these quantities, in the Bernoulli HIBP and general spike and slab HIBP settings, which allows for exact sampling and simpler practical implementation. We then extend these results to the more complex setting of hierarchies of general HIBP (HHIBP). The generality of our framework allows one to recognize important structure that may otherwise be masked in the Bernoulli setting, and involves characterizations via dynamic mixed Poisson random count matrices. Our analysis shows that the standard choice of hierarchical Beta processes for modeling across group sharing is not ideal in the classic Bernoulli HIBP setting proposed by Thibaux and Jordan (2007), or other spike and slab HIBP settings, and we thus indicate tractable alternative priors.

Spiking neural network models characterize the emergent collective dynamics of circuits of biological neurons and help engineer neuro-inspired solutions across fields. Most dynamical systems' models of spiking neural networks typically exhibit one of two major types of interactions: First, the response of a neuron's state variable to incoming pulse signals (spikes) may be additive and independent of its current state. Second, the response may depend on the current neuron's state and multiply a function of the state variable. Here we reveal that spiking neural network models with additive coupling are equivalent to models with multiplicative coupling for simultaneously modified intrinsic neuron time evolution. As a consequence, the same collective dynamics can be attained by state-dependent multiplicative and constant (state-independent) additive coupling. Such a mapping enables the transfer of theoretical insights between spiking neural network models with different types of interaction mechanisms as well as simpler and more effective engineering applications.

This paper addresses the deconvolution problem of estimating a square-integrable probability density from observations contaminated with additive measurement errors having a known density. The estimator begins with a density estimate of the contaminated observations and minimizes a reconstruction error penalized by an integrated squared $m$-th derivative. Theory for deconvolution has mainly focused on kernel- or wavelet-based techniques, but other methods including spline-based techniques and this smoothness-penalized estimator have been found to outperform kernel methods in simulation studies. This paper fills in some of these gaps by establishing asymptotic guarantees for the smoothness-penalized approach. Consistency is established in mean integrated squared error, and rates of convergence are derived for Gaussian, Cauchy, and Laplace error densities, attaining some lower bounds already in the literature. The assumptions are weak for most results; the estimator can be used with a broader class of error densities than the deconvoluting kernel. Our application example estimates the density of the mean cytotoxicity of certain bacterial isolates under random sampling; this mean cytotoxicity can only be measured experimentally with additive error, leading to the deconvolution problem. We also describe a method for approximating the solution by a cubic spline, which reduces to a quadratic program.

Mixture distributions with dynamic weights are an efficient way of modeling loss data characterized by heavy tails. However, maximum likelihood estimation of this family of models is difficult, mostly because of the need to evaluate numerically an intractable normalizing constant. In such a setup, simulation-based estimation methods are an appealing alternative. The approximate maximum likelihood estimation (AMLE) approach is employed. It is a general method that can be applied to mixtures with any component densities, as long as simulation is feasible. The focus is on the dynamic lognormal-generalized Pareto distribution, and the Cram\'er - von Mises distance is used to measure the discrepancy between observed and simulated samples. After deriving the theoretical properties of the estimators, a hybrid procedure is developed, where standard maximum likelihood is first employed to determine the bounds of the uniform priors required as input for AMLE. Simulation experiments and two real-data applications suggest that this approach yields a major improvement with respect to standard maximum likelihood estimation.

Communication over a classical multiple-access channel (MAC) with entanglement resources is considered, whereby two transmitters share entanglement resources a priori before communication begins. Leditzki et al. (2020) presented an example of a classical MAC, defined in terms of a pseudo telepathy game, such that the sum rate with entangled transmitters is strictly higher than the best achievable sum rate without such resources. Here, we determine the capacity region for the general MAC with entangled transmitters, and show that the previous result can be obtained as a special case. Furthermore, it has long been known that the capacity region of the classical MAC under a message-average error criterion can be strictly larger than with a maximal error criterion (Dueck, 1978). We observe that given entanglement resources, the regions coincide.

Cognitive Diagnosis Models (CDMs) are a powerful statistical and psychometric tool for researchers and practitioners to learn fine-grained diagnostic information about respondents' latent attributes. There has been a growing interest in the use of CDMs for polytomous response data, as more and more items with multiple response options become widely used. Similar to many latent variable models, the identifiability of CDMs is critical for accurate parameter estimation and valid statistical inference. However, the existing identifiability results are primarily focused on binary response models and have not adequately addressed the identifiability of CDMs with polytomous responses. This paper addresses this gap by presenting sufficient and necessary conditions for the identifiability of the widely used DINA model with polytomous responses, with the aim to provide a comprehensive understanding of the identifiability of CDMs with polytomous responses and to inform future research in this field.

The dynamics of neuron populations during diverse tasks often evolve on low-dimensional manifolds. However, it remains challenging to discern the contributions of geometry and dynamics for encoding relevant behavioural variables. Here, we introduce an unsupervised geometric deep learning framework for representing non-linear dynamical systems based on statistical distributions of local phase portrait features. Our method provides robust geometry-aware or geometry-agnostic representations for the unbiased comparison of dynamics based on measured trajectories. We demonstrate that our statistical representation can generalise across neural network instances to discriminate computational mechanisms, obtain interpretable embeddings of neural dynamics in a primate reaching task with geometric correspondence to hand kinematics, and develop a decoding algorithm with state-of-the-art accuracy. Our results highlight the importance of using the intrinsic manifold structure over temporal information to develop better decoding algorithms and assimilate data across experiments.

The rapid recent progress in machine learning (ML) has raised a number of scientific questions that challenge the longstanding dogma of the field. One of the most important riddles is the good empirical generalization of overparameterized models. Overparameterized models are excessively complex with respect to the size of the training dataset, which results in them perfectly fitting (i.e., interpolating) the training data, which is usually noisy. Such interpolation of noisy data is traditionally associated with detrimental overfitting, and yet a wide range of interpolating models -- from simple linear models to deep neural networks -- have recently been observed to generalize extremely well on fresh test data. Indeed, the recently discovered double descent phenomenon has revealed that highly overparameterized models often improve over the best underparameterized model in test performance. Understanding learning in this overparameterized regime requires new theory and foundational empirical studies, even for the simplest case of the linear model. The underpinnings of this understanding have been laid in very recent analyses of overparameterized linear regression and related statistical learning tasks, which resulted in precise analytic characterizations of double descent. This paper provides a succinct overview of this emerging theory of overparameterized ML (henceforth abbreviated as TOPML) that explains these recent findings through a statistical signal processing perspective. We emphasize the unique aspects that define the TOPML research area as a subfield of modern ML theory and outline interesting open questions that remain.

Small data challenges have emerged in many learning problems, since the success of deep neural networks often relies on the availability of a huge amount of labeled data that is expensive to collect. To address it, many efforts have been made on training complex models with small data in an unsupervised and semi-supervised fashion. In this paper, we will review the recent progresses on these two major categories of methods. A wide spectrum of small data models will be categorized in a big picture, where we will show how they interplay with each other to motivate explorations of new ideas. We will review the criteria of learning the transformation equivariant, disentangled, self-supervised and semi-supervised representations, which underpin the foundations of recent developments. Many instantiations of unsupervised and semi-supervised generative models have been developed on the basis of these criteria, greatly expanding the territory of existing autoencoders, generative adversarial nets (GANs) and other deep networks by exploring the distribution of unlabeled data for more powerful representations. While we focus on the unsupervised and semi-supervised methods, we will also provide a broader review of other emerging topics, from unsupervised and semi-supervised domain adaptation to the fundamental roles of transformation equivariance and invariance in training a wide spectrum of deep networks. It is impossible for us to write an exclusive encyclopedia to include all related works. Instead, we aim at exploring the main ideas, principles and methods in this area to reveal where we are heading on the journey towards addressing the small data challenges in this big data era.

With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.