Out-of-distribution (OOD) detection is essential for determining when a supervised model encounters inputs that differ meaningfully from its training distribution. While widely studied in classification, OOD detection for regression and survival analysis remains limited due to the absence of discrete labels and the challenge of quantifying predictive uncertainty. We introduce a framework for OOD detection that is simultaneously model aware and subspace aware, and that embeds variable prioritization directly into the detection step. The method uses the fitted predictor to construct localized neighborhoods around each test case that emphasize the features driving the model's learned relationship and downweight directions that are less relevant to prediction. It produces OOD scores without relying on global distance metrics or estimating the full feature density. The framework is applicable across outcome types, and in our implementation we use random forests, where the rule structure yields transparent neighborhoods and effective scoring. Experiments on synthetic and real data benchmarks designed to isolate functional shifts show consistent improvements over existing methods. We further demonstrate the approach in an esophageal cancer survival study, where distribution shifts related to lymphadenectomy identify patterns relevant to surgical guidelines.

Mixture models are widely used in modeling heterogeneous data populations. A standard approach of mixture modeling is to assume that the mixture component takes a parametric kernel form, while the flexibility of the model can be obtained by using a large or possibly unbounded number of such parametric kernels. In many applications, making parametric assumptions on the latent subpopulation distributions may be unrealistic, which motivates the need for nonparametric modeling of the mixture components themselves. In this paper we study finite mixtures with nonparametric mixture components, using a Bayesian nonparametric modeling approach. In particular, it is assumed that the data population is generated according to a finite mixture of latent component distributions, where each component is endowed with a Bayesian nonparametric prior such as the Dirichlet process mixture. We present conditions under which the individual mixture component's distributions can be identified, and establish posterior contraction behavior for the data population's density, as well as densities of the latent mixture components. We develop an efficient MCMC algorithm for posterior inference and demonstrate via simulation studies and real-world data illustrations that it is possible to efficiently learn complex distributions for the latent subpopulations. In theory, the posterior contraction rate of the component densities is nearly polynomial, which is a significant improvement over the logarithm convergence rate of estimating mixing measures via deconvolution.

While joint pruning--quantization is theoretically superior to sequential application, current joint methods rely on auxiliary procedures outside the training loop for finding compression parameters. This reliance adds engineering complexity and hyperparameter tuning, while also lacking a direct data-driven gradient signal, which might result in sub-optimal compression. In this paper, we introduce CoDeQ, a simple, fully differentiable method for joint pruning--quantization. Our approach builds on a key observation: the dead-zone of a scalar quantizer is equivalent to magnitude pruning, and can be used to induce sparsity directly within the quantization operator. Concretely, we parameterize the dead-zone width and learn it via backpropagation, alongside the quantization parameters. This design provides explicit control of sparsity, regularized by a single global hyperparameter, while decoupling sparsity selection from bit-width selection. The result is a method for Compression with Dead-zone Quantizer (CoDeQ) that supports both fixed-precision and mixed-precision quantization (controlled by an optional second hyperparameter). It simultaneously determines the sparsity pattern and quantization parameters in a single end-to-end optimization. Consequently, CoDeQ does not require any auxiliary procedures, making the method architecture-agnostic and straightforward to implement. On ImageNet with ResNet-18, CoDeQ reduces bit operations to ~5% while maintaining close to full precision accuracy in both fixed and mixed-precision regimes.

Identifying and comparing topological features, particularly cycles, across different topological objects remains a fundamental challenge in persistent homology and topological data analysis. This work introduces a novel framework for constructing cycle communities through two complementary approaches. First, a dendrogram-based methodology leverages merge-tree algorithms to construct hierarchical representations of homology classes from persistence intervals. The Wasserstein distance on merge trees is introduced as a metric for comparing dendrograms, establishing connections to hierarchical clustering frameworks. Through simulation studies, the discriminative power of dendrogram representations for identifying cycle communities is demonstrated. Second, an extension of Stratified Gradient Sampling simultaneously learns multiple filter functions that yield cycle barycenter functions capable of faithfully reconstructing distinct sets of cycles. The set of cycles each filter function can reconstruct constitutes cycle communities that are non-overlapping and partition the space of all cycles. Together, these approaches transform the problem of cycle matching into both a hierarchical clustering and topological optimization framework, providing principled methods to identify similar topological structures both within and across groups of topological objects.

We develop a rigorous walk-forward validation framework for algorithmic trading designed to mitigate overfitting and lookahead bias. Our methodology combines interpretable hypothesis-driven signal generation with reinforcement learning and strict out-of-sample testing. The framework enforces strict information set discipline, employs rolling window validation across 34 independent test periods, maintains complete interpretability through natural language hypothesis explanations, and incorporates realistic transaction costs and position constraints. Validating five market microstructure patterns across 100 US equities from 2015 to 2024, the system yields modest annualized returns (0.55%, Sharpe ratio 0.33) with exceptional downside protection (maximum drawdown -2.76%) and market-neutral characteristics (beta = 0.058). Performance exhibits strong regime dependence, generating positive returns during high-volatility periods (0.60% quarterly, 2020-2024) while underperforming in stable markets (-0.16%, 2015-2019). We report statistically insignificant aggregate results (p-value 0.34) to demonstrate a reproducible, honest validation protocol that prioritizes interpretability and extends naturally to advanced hypothesis generators, including large language models. The key empirical finding reveals that daily OHLCV-based microstructure signals require elevated information arrival and trading activity to function effectively. The framework provides complete mathematical specifications and open-source implementation, establishing a template for rigorous trading system evaluation that addresses the reproducibility crisis in quantitative finance research. For researchers, practitioners, and regulators, this work demonstrates that interpretable algorithmic trading strategies can be rigorously validated without sacrificing transparency or regulatory compliance.

This study evaluates thresholds for removing singular values from singular value decomposition-based low-rank approximations of deep neural network weight matrices. Each weight matrix is modeled as the sum of signal and noise matrices. The low-rank approximation is obtained by removing noise-related singular values using a threshold based on random matrix theory. To assess the adequacy of this threshold, we propose an evaluation metric based on the cosine similarity between the singular vectors of the signal and original weight matrices. The proposed metric is used in numerical experiments to compare two threshold estimation methods.

Linear Autoencoders (LAEs) have shown strong performance in state-of-the-art recommender systems. However, this success remains largely empirical, with limited theoretical understanding. In this paper, we investigate the generalizability -- a theoretical measure of model performance in statistical learning -- of multivariate linear regression and LAEs. We first propose a PAC-Bayes bound for multivariate linear regression, extending the earlier bound for single-output linear regression by Shalaeva et al., and establish sufficient conditions for its convergence. We then show that LAEs, when evaluated under a relaxed mean squared error, can be interpreted as constrained multivariate linear regression models on bounded data, to which our bound adapts. Furthermore, we develop theoretical methods to improve the computational efficiency of optimizing the LAE bound, enabling its practical evaluation on large models and real-world datasets. Experimental results demonstrate that our bound is tight and correlates well with practical ranking metrics such as Recall@K and NDCG@K.

Neural population activity often exhibits regime-dependent non-stationarity in the form of switching dynamics. Learning accurate switching dynamical system models can reveal how behavior is encoded in neural activity. Existing switching approaches have primarily focused on learning models from a single neural modality, either continuous Gaussian signals or discrete Poisson signals. However, multiple neural modalities are often recorded simultaneously to measure different spatiotemporal scales of brain activity, and all these modalities can encode behavior. Moreover, regime labels are typically unavailable in training data, posing a significant challenge for learning models of regime-dependent switching dynamics. To address these challenges, we develop a novel unsupervised learning algorithm that learns the parameters of switching multiscale dynamical system models using only multiscale neural observations. We demonstrate our method using both simulations and two distinct experimental datasets with multimodal spike-LFP observations during different motor tasks. We find that our switching multiscale dynamical system models more accurately decode behavior than switching single-scale dynamical models, showing the success of multiscale neural fusion. Further, our models outperform stationary multiscale models, illustrating the importance of tracking regime-dependent non-stationarity in multimodal neural data. The developed unsupervised learning framework enables more accurate modeling of complex multiscale neural dynamics by leveraging information in multimodal recordings while incorporating regime switches. This approach holds promise for improving the performance and robustness of brain-computer interfaces over time and for advancing our understanding of the neural basis of behavior.

Markov Chain Monte Carlo (MCMC), Laplace approximation (LA) and variational inference (VI) methods are popular approaches to Bayesian inference, each with trade-offs between computational cost and accuracy. However, a theoretical understanding of these differences is missing, particularly when both the sample size $n$ and the dimension $d$ are large. LA and Gaussian VI are justified by Bernstein-von Mises (BvM) theorems, and recent work has derived the characteristic condition $n\gg d^2$ for their validity, improving over the condition $n\gg d^3$. In this paper, we show for linear, logistic and Poisson regression that for $n\gtrsim d$, MCMC attains the same complexity scaling in $n$, $d$ as first-order optimization algorithms, up to sub-polynomial factors. Thus MCMC is competitive with LA and Gaussian VI in complexity, under a scaling between $n$ and $d$ more general than BvM regimes. Our complexities apply to appropriately scaled priors that are not necessarily Gaussian-tailed, including Student-$t$ and flat priors, with log-posteriors that are not necessarily globally concave or gradient-Lipschitz.

We present a framework using variational inference with normalizing flows (VI-NFs) to generate proposals of reversible jump Markov chain Monte Carlo (RJMCMC) for efficient trans-dimensional Bayesian inference. Unlike transport reversible jump methods relying on forward KL minimization with pilot MCMC samples, our approach minimizes the reverse KL divergence which requires only samples from a base distribution, eliminating costly target sampling. The method employs RealNVP-based flows to learn model-specific transport maps, enabling construction of both between-model and within-model proposals. Our framework provides accurate marginal likelihood estimates from the variational approximation. This facilitates efficient model comparison and proposal adaptation in RJMCMC. Experiments on illustrative example, factor analysis and variable selection tasks in linear regression show that TRJ designed by VI-NFs achieves faster mixing and more efficient model space exploration compared to existing baselines. The proposed algorithm can be extended to conditional flows for amortized vairiational inference across models. Code is available at https://github.com/YinPingping111/TRJ_VINFs.