As a nonlinear dimension reduction technique, the diffusion map (DM) has been widely used. In DM, kernels play an important role for capturing the nonlinear relationship of data. However, only symmetric kernels can be used now, which prevents the use of DM in directed graphs, trophic networks, and other real-world scenarios where the intrinsic and extrinsic geometries in data are asymmetric. A promising technique is the magnetic transform which converts an asymmetric matrix to a Hermitian one. However, we are facing essential problems, including how diffusion distance could be preserved and how divergence could be avoided during diffusion process. Via theoretical proof, we successfully establish a diffusion representation framework with the magnetic transform, named MagDM. The effectiveness and robustness for dealing data endowed with asymmetric proximity are demonstrated on three synthetic datasets and two trophic networks.

Communication compression is a common technique in distributed optimization that can alleviate communication overhead by transmitting compressed gradients and model parameters. However, compression can introduce information distortion, which slows down convergence and incurs more communication rounds to achieve desired solutions. Given the trade-off between lower per-round communication costs and additional rounds of communication, it is unclear whether communication compression reduces the total communication cost. This paper explores the conditions under which unbiased compression, a widely used form of compression, can reduce the total communication cost, as well as the extent to which it can do so. To this end, we present the first theoretical formulation for characterizing the total communication cost in distributed optimization with unbiased compressors. We demonstrate that unbiased compression alone does not necessarily save the total communication cost, but this outcome can be achieved if the compressors used by all workers are further assumed independent. We establish lower bounds on the communication rounds required by algorithms using independent unbiased compressors to minimize smooth convex functions and show that these lower bounds are tight by refining the analysis for ADIANA. Our results reveal that using independent unbiased compression can reduce the total communication cost by a factor of up to Θ( p min{n,κ}) when all local smoothness constants are constrained by a common upper bound, where nis the number of workers and κis the condition number of the functions being minimized. These theoretical findings are supported by experimental results.

The relationship between perception and inference, as postulated by Helmholtz in the 19th century, is paralleled in modern machine learning by generative models like Variational Autoencoders (VAEs) and their hierarchical variants. Here, we evaluate the role of hierarchical inference and its alignment with brain function in the domain of motion perception. We first introduce a novel synthetic data framework, Retinal Optic Flow Learning (ROFL), which enables control over motion statistics and their causes. We then present a new hierarchical VAE and test it against alternative models on two downstream tasks: (i) predicting ground truth causes of retinal optic flow (e.g., self-motion); and (ii) predicting the responses of neurons in the motion processing pathway of primates. We manipulate the model architectures (hierarchical versus non-hierarchical), loss functions, and the causal structure of the motion stimuli.

We present a systematic recipe for translating ReLU approximation results to softmax Transformers1. Given a constructive ReLU approximator for a target, we construct an explicit softmax transformer with the same accuracy. The recipe applies to many common approximation targets and yields quantitative resource bounds beyond universal approximation statements. This matters because broad Universal Approximation Properties (UAP) still dominate Transformer approximation theory. For softmax Transformer, many universality results provide explicit constructions and quantitative resource bounds (e.g., parameters, depth, width...etc) [Yun et al., 2020, Kajitsuka and Sato, 2023, Takakura and Suzuki, 2023, Jiang and Li, 2024, Hu et al., 2025,

Nonlinear dynamical systems with regime transitions are typically described by ordinary differential equations with jumping parameters parameters. Traditional methods often treat change-point detection and parameter estimation as separate tasks, ignoring the inherent coupling between them. To address this, we propose residual-loss anomaly analysis of physics-informed neural networks, a unified framework that leverages dynamical consistency within the physics-informed learning paradigm. This approach jointly infers piecewise parameters and transition points under a single set of constraints. The method follows a two-stage strategy: First, local physical residuals are analyzed through overlapping subinterval decomposition. When a subinterval spans a true transition point, the residual exhibits a distinct structural elevation in noise-free conditions, which has a non-zero lower bound, enabling effective localization of potential transition intervals. Second, within our framework, change-point locations and piecewise parameters are integrated into a unified physical loss function for joint optimization, enabling simultaneous identification. Experiments on benchmark nonlinear dynamical systems, including Malthusian and logistic growth models, Van der Pol oscillator, Lotka-Volterra model and Lorenz system, demonstrate that the proposed method outperforms traditional decoupled approaches in both change-point localization and parameter estimation accuracy. This study provides an efficient, unified solution for structurally coupled inverse problems in nonlinear dynamical systems with regime switching.

Sparse Optimal Scoring (SOS) reformulates linear discriminant analysis to enable feature selection through elastic net regularization, making it well-suited for high-dimensional settings where the number of features exceeds observations. Most existing SOS methods use deflation-based strategies that compute discriminant vectors sequentially, which can propagate errors and produce suboptimal solutions. We propose a novel approach that estimates all discriminant vectors simultaneously under an explicit global orthogonality constraint, which we call Deflation-Free Sparse Optimal Scoring (DFSOS). DFSOS combines Bregman iteration with orthogonality-constrained optimization, decomposing the problem into tractable subproblems for scoring vectors, discriminant vectors, and orthogonality enforcement. We establish convergence to stationary points of the augmented Lagrangian under mild conditions. Extensive experiments using synthetic data and real-world time series data demonstrate that DFSOS achieves classification accuracy comparable to or better than existing deflation-based methods. These results indicate that deflation-free approaches offer a robust and effective framework for sparse discriminant analysis in high-dimensional problems.