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Normalizing flows, diffusion normalizing flows and variational autoencoders are powerful generative models. In this paper, we provide a unified framework to handle these approaches via Markov chains. Indeed, we consider stochastic normalizing flows as pair of Markov chains fulfilling some properties and show that many state-of-the-art models for data generation fit into this framework. The Markov chains point of view enables us to couple both deterministic layers as invertible neural networks and stochastic layers as Metropolis-Hasting layers, Langevin layers and variational autoencoders in a mathematically sound way. Besides layers with densities as Langevin layers, diffusion layers or variational autoencoders, also layers having no densities as deterministic layers or Metropolis-Hasting layers can be handled. Hence our framework establishes a useful mathematical tool to combine the various approaches.

In this paper we consider the spatial semi-discretization of conservative PDEs. Such finite dimensional approximations of infinite dimensional dynamical systems can be described as flows in suitable matrix spaces, which in turn leads to the need to solve polynomial matrix equations, a classical and important topic both in theoretical and in applied mathematics. Solving numerically these equations is challenging due to the presence of several conservation laws which our finite models incorporate and which must be retained while integrating the equations of motion. In the last thirty years, the theory of geometric integration has provided a variety of techniques to tackle this problem. These numerical methods require to solve both direct and inverse problems in matrix spaces. We present two algorithms to solve a cubic matrix equation arising in the geometric integration of isospectral flows. This type of ODEs includes finite models of ideal hydrodynamics, plasma dynamics, and spin particles, which we use as test problems for our algorithms.

Hypercontractivity is one of the most powerful tools in Boolean function analysis. Originally studied over the discrete hypercube, recent years have seen increasing interest in extensions to settings like the $p$-biased cube, slice, or Grassmannian, where variants of hypercontractivity have found a number of breakthrough applications including the resolution of Khot's 2-2 Games Conjecture (Khot, Minzer, Safra FOCS 2018). In this work, we develop a new theory of hypercontractivity on high dimensional expanders (HDX), an important class of expanding complexes that has recently seen similarly impressive applications in both coding theory and approximate sampling. Our results lead to a new understanding of the structure of Boolean functions on HDX, including a tight analog of the KKL Theorem and a new characterization of non-expanding sets. Unlike previous settings satisfying hypercontractivity, HDX can be asymmetric, sparse, and very far from products, which makes the application of traditional proof techniques challenging. We handle these barriers with the introduction of two new tools of independent interest: a new explicit combinatorial Fourier basis for HDX that behaves well under restriction, and a new local-to-global method for analyzing higher moments. Interestingly, unlike analogous second moment methods that apply equally across all types of expanding complexes, our tools rely inherently on simplicial structure. This suggests a new distinction among high dimensional expanders based upon their behavior beyond the second moment.

An $n$-dimensional source with memory is observed by $K$ isolated encoders via parallel channels, who compress their observations to transmit to the decoder via noiseless rate-constrained links while leveraging their memory of the past. At each time instant, the decoder receives $K$ new codewords from the observers, combines them with the past received codewords, and produces a minimum-distortion estimate of the latest block of $n$ source symbols. This scenario extends the classical one-shot CEO problem to multiple rounds of communication with communicators maintaining the memory of the past. We extend the Berger-Tung inner and outer bounds to the scenario with inter-block memory, showing that the minimum asymptotically (as $n \to \infty$) achievable sum rate required to achieve a target distortion is bounded by minimal directed mutual information problems. For the Gauss-Markov source observed via $K$ parallel AWGN channels, we show that the inner bound is tight and solve the corresponding minimal directed mutual information problem, thereby establishing the minimum asymptotically achievable sum rate. Finally, we explicitly bound the rate loss due to a lack of communication among the observers; that bound is attained with equality in the case of identical observation channels. The general coding theorem is proved via a new nonasymptotic bound that uses stochastic likelihood coders and whose asymptotic analysis yields an extension of the Berger-Tung inner bound to the causal setting. The analysis of the Gaussian case is facilitated by reversing the channels of the observers.

Reconfigurable intelligent surfaces (RISs) represent a promising candidate for sixth-generation (6G) wireless networks, as the RIS technology provides a new solution to control the propagation channel in order to improve the efficiency of a wireless link through enhancing the received signal power. In this paper, we propose RIS-assisted receive quadrature space-shift keying (RIS-RQSSK), which enhances the spectral efficiency of an RIS-based index modulation (IM) system by using the real and imaginary dimensions independently for the purpose of IM. Therefore, the error rate performance of the system is improved as all RIS elements reflect the incident transmit signal toward both selected receive antennas. At the receiver, a low-complexity but effective greedy detector (GD) can be employed which determines the maximum energy per dimension at the receive antennas. A max-min optimization problem is defined to maximize the received signal-to-noise ratio (SNR) components at both selected receive antennas; an analytical solution is provided based on Lagrange duality. In particular, the multi-variable optimization problem is shown to reduce to the solution of a single-variable equation, which results in a very simple design procedure. In addition, we investigate the average bit error probability (ABEP) of the proposed RIS-RQSSK system and derive a closed-form approximate upper bound on the ABEP. We also provide extensive numerical simulations to validate our derivations. Numerical results show that the proposed RIS-RQSSK scheme substantially outperforms recent prominent benchmark schemes. This enhancement considerably increases with an increasing number of receive antennas.

Despite significant economic and ecological effects, a higher level of renewable energy generation leads to increased uncertainty and variability in power injections, thus compromising grid reliability. In order to improve power grid security, we investigate a joint chance-constrained (CC) direct current (DC) optimal power flow (OPF) problem. The problem aims to find economically optimal power generation while guaranteeing that all power generation, line flows, and voltages simultaneously remain within their bounds with a pre-defined probability. Unfortunately, the problem is computationally intractable even if the distribution of renewables fluctuations is specified. Moreover, existing approximate solutions to the joint CC OPF problem are overly conservative, and therefore have less value for the operational practice. This paper proposes an importance sampling approach to the CC DC OPF problem, which yields better complexity and accuracy than current state-of-the-art methods. The algorithm efficiently reduces the number of scenarios by generating and using only the most important of them, thus enabling real-time solutions for test cases with up to several hundred buses.

Discrete-event systems usually consist of discrete states and transitions between them caused by spontaneous occurrences of labelled (aka partially-observed) events. Due to the partially-observed feature, fundamental properties therein could be classified into two categories: state/event-inference-based properties (e.g., strong detectability, diagnosability, and predictability) and state-concealment-based properties (e.g., opacity). Intuitively, the former category describes whether one can use observed output sequences to infer the current and subsequent states, past occurrences of faulty events, or future certain occurrences of faulty events; while the latter describes whether one cannot use observed output sequences to infer whether some secret states have been visited (that is, whether the DES can conceal the status that its secret states have been visited). Over the past two decades these properties were studied separately using different methods. In this review article, for labeled finite-state automata, a unified concurrent-composition method is shown to verify all above inference-based properties and concealment-based properties, resulting in a unified mathematical framework for the two categories of properties. In addition, compared with the previous methods in the literature, the concurrent-composition method does not depend on assumptions and is more efficient.

We propose a general and scalable approximate sampling strategy for probabilistic models with discrete variables. Our approach uses gradients of the likelihood function with respect to its discrete inputs to propose updates in a Metropolis-Hastings sampler. We show empirically that this approach outperforms generic samplers in a number of difficult settings including Ising models, Potts models, restricted Boltzmann machines, and factorial hidden Markov models. We also demonstrate the use of our improved sampler for training deep energy-based models on high dimensional discrete data. This approach outperforms variational auto-encoders and existing energy-based models. Finally, we give bounds showing that our approach is near-optimal in the class of samplers which propose local updates.

Temporal action proposal generation aims to estimate temporal intervals of actions in untrimmed videos, which is a challenging yet important task in the video understanding field. The proposals generated by current methods still suffer from inaccurate temporal boundaries and inferior confidence used for retrieval owing to the lack of efficient temporal modeling and effective boundary context utilization. In this paper, we propose Temporal Context Aggregation Network (TCANet) to generate high-quality action proposals through "local and global" temporal context aggregation and complementary as well as progressive boundary refinement. Specifically, we first design a Local-Global Temporal Encoder (LGTE), which adopts the channel grouping strategy to efficiently encode both "local and global" temporal inter-dependencies. Furthermore, both the boundary and internal context of proposals are adopted for frame-level and segment-level boundary regressions, respectively. Temporal Boundary Regressor (TBR) is designed to combine these two regression granularities in an end-to-end fashion, which achieves the precise boundaries and reliable confidence of proposals through progressive refinement. Extensive experiments are conducted on three challenging datasets: HACS, ActivityNet-v1.3, and THUMOS-14, where TCANet can generate proposals with high precision and recall. By combining with the existing action classifier, TCANet can obtain remarkable temporal action detection performance compared with other methods. Not surprisingly, the proposed TCANet won the 1$^{st}$ place in the CVPR 2020 - HACS challenge leaderboard on temporal action localization task.

This paper addresses the problem of formally verifying desirable properties of neural networks, i.e., obtaining provable guarantees that neural networks satisfy specifications relating their inputs and outputs (robustness to bounded norm adversarial perturbations, for example). Most previous work on this topic was limited in its applicability by the size of the network, network architecture and the complexity of properties to be verified. In contrast, our framework applies to a general class of activation functions and specifications on neural network inputs and outputs. We formulate verification as an optimization problem (seeking to find the largest violation of the specification) and solve a Lagrangian relaxation of the optimization problem to obtain an upper bound on the worst case violation of the specification being verified. Our approach is anytime i.e. it can be stopped at any time and a valid bound on the maximum violation can be obtained. We develop specialized verification algorithms with provable tightness guarantees under special assumptions and demonstrate the practical significance of our general verification approach on a variety of verification tasks.