The sampling of probability distributions specified up to a normalization constant is an important problem in both machine learning and statistical mechanics. While classical stochastic sampling methods such as Markov Chain Monte Carlo (MCMC) or Langevin Dynamics (LD) can suffer from slow mixing times there is a growing interest in using normalizing flows in order to learn the transformation of a simple prior distribution to the given target distribution. Here we propose a generalized and combined approach to sample target densities: Stochastic Normalizing Flows (SNF) - an arbitrary sequence of deterministic invertible functions and stochastic sampling blocks. We show that stochasticity overcomes expressivity limitations of normalizing flows resulting from the invertibility constraint, whereas trainable transformations between sampling steps improve efficiency of pure MCMC/LD along the flow. By invoking ideas from non-equilibrium statistical mechanics we derive an efficient training procedure by which both the sampler's and the flow's parameters can be optimized end-to-end, and by which we can compute exact importance weights without having to marginalize out the randomness of the stochastic blocks. We illustrate the representational power, sampling efficiency and asymptotic correctness of SNFs on several benchmarks including applications to sampling molecular systems in equilibrium.

More complex data consisting of measurements across multiple modalities can be organized as higher dimensional data arrays, or higher order tensors.

Neural Information Processing Systems http://nips.cc/

Interestingly, networks with a large number of gating units behave similarly to standard ReLU architectures.

Despite the striking successes of deep neural networks trained with gradient-based optimization, these methods differ fundamentally from their biological counterparts. This gap raises key questions about how nature achieves robust, sample-efficient learning at minimal energy costs and solves the credit-assignment problem without backpropagation. We take a step toward bridging contemporary AI and computational neuroscience by studying how neural dynamics can support fully local, distributed learning that scales to simple machine-learning benchmarks. Using tools from statistical mechanics, we identify conditions for the emergence of robust dynamical attractors in random asymmetric recurrent networks. We derive a closed-form expression for the number of fixed points as a function of self-coupling strength, and we reveal a phase transition in their structure: below a critical self-coupling, isolated fixed points coexist with exponentially many narrow clusters showing the overlap-gap property; above it, subdominant yet dense and extensive clusters appear. These fixed points become accessible, including to a simple asynchronous dynamical rule, after an algorithm-dependent self-coupling threshold. Building on this analysis, we propose a biologically plausible algorithm for supervised learning with any binary recurrent network. Inputs are mapped to fixed points of the dynamics, by relaxing under transient external stimuli and stabilizing the resulting configurations via local plasticity. We show that our algorithm can learn an entangled version of MNIST, leverages depth to develop hierarchical representations and increase hetero-association capacity, and is applicable to several architectures. Finally, we highlight the strong connection between algorithm performance and the unveiled phase transition, and we suggest a cortex-inspired alternative to self-couplings for its emergence.

Complex networks have become essential tools for understanding diverse phenomena in social systems, traffic systems, biomolecular systems, and financial systems. Identifying critical nodes is a central theme in contemporary research, serving as a vital bridge between theoretical foundations and practical applications. Nevertheless, the intrinsic complexity and structural heterogeneity characterizing real-world networks, with particular emphasis on dynamic and higher-order networks, present substantial obstacles to the development of universal frameworks for critical node identification. This paper provides a comprehensive review of critical node identification techniques, categorizing them into seven main classes: centrality, critical nodes deletion problem, influence maximization, network control, artificial intelligence, higher-order and dynamic methods. Our review bridges the gaps in existing surveys by systematically classifying methods based on their methodological foundations and practical implications, and by highlighting their strengths, limitations, and applicability across different network types. Our work enhances the understanding of critical node research by identifying key challenges, such as algorithmic universality, real-time evaluation in dynamic networks, analysis of higher-order structures, and computational efficiency in large-scale networks. The structured synthesis consolidates current progress and highlights open questions, particularly in modeling temporal dynamics, advancing efficient algorithms, integrating machine learning approaches, and developing scalable and interpretable metrics for complex systems.

Hierarchical structures, which include multiple levels, are prevalent in statistical and machine-learning models as well as physical systems. Extending the foundational result that the maximum entropy distribution under mean constraints is given by the exponential Gibbs-Boltzmann form, we introduce the framework of "hierarchical maximum entropy" to address these multilevel models. We demonstrate that Pareto optimal distributions, which maximize entropies across all levels of hierarchical transformations, can be obtained via renormalization-group procedures from theoretical physics. This is achieved by formulating multilevel extensions of the Gibbs variational principle and the Donsker-Varadhan variational representation of entropy. Moreover, we explore settings with hierarchical invariances that significantly simplify the renormalization-group procedures, enhancing computational efficiency: quadratic modular loss functions, logarithmic loss functions, and nearest-neighbor loss functions. This is accomplished through the introduction of the concept of parameter flows, which serves as an analog to renormalization flows in renormalization group theory. This work connects ideas from probability theory, information theory, and statistical mechanics.

Interestingly, networks with a large number of gating units behave similarly to standard ReLU architectures.