Recent years have witnessed significant advancements in machine learning methods on graphs. However, transferring knowledge effectively from one graph to another remains a critical challenge. This highlights the need for algorithms capable of applying information extracted from a source graph to an unlabeled target graph, a task known as unsupervised graph domain adaptation (GDA). One key difficulty in unsupervised GDA is conditional shift, which hinders transferability. In this paper, we show that conditional shift can be observed only if there exists local dependencies among node features. To support this claim, we perform a rigorous analysis and also further provide generalization bounds of GDA when dependent node features are modeled using markov chains. Guided by the theoretical findings, we propose to improve GDA by decorrelating node features, which can be specifically implemented through decorrelated GCN layers and graph transformer layers. Our experimental results demonstrate the effectiveness of this approach, showing not only substantial performance enhancements over baseline GDA methods but also clear visualizations of small intra-class distances in the learned representations. Our code is available at https://github.com/TechnologyAiGroup/DFT

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.

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.

Graph Convolutional Networks (GCNs) have become a standard approach for semi-supervised node classification, yet practitioners lack clear guidance on when GCNs provide meaningful improvements over simpler baselines. We present a diagnostic study using the Amazon Computers co-purchase data to understand when and why GCNs help. Through systematic experiments with simulated label scarcity, feature ablation, and per-class analysis, we find that GCN performance depends critically on the interaction between graph homophily and feature quality. GCNs provide the largest gains under extreme label scarcity, where they leverage neighborhood structure to compensate for limited supervision. Surprisingly, GCNs can match their original performance even when node features are replaced with random noise, suggesting that structure alone carries sufficient signal on highly homophilous graphs. However, GCNs hurt performance when homophily is low and features are already strong, as noisy neighbors corrupt good predictions. Our quadrant analysis reveals that GCNs help in three of four conditions and only hurt when low homophily meets strong features. These findings offer practical guidance for practitioners deciding whether to adopt graph-based methods.

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.

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.

Machine learning (ML) methods are highly flexible, but their ability to approximate the true data-generating process is fundamentally constrained by finite samples. We characterize a universal lower bound, the Limits-to-Learning Gap (LLG), quantifying the unavoidable discrepancy between a model's empirical fit and the population benchmark. Recovering the true population $R^2$, therefore, requires correcting observed predictive performance by this bound. Using a broad set of variables, including excess returns, yields, credit spreads, and valuation ratios, we find that the implied LLGs are large. This indicates that standard ML approaches can substantially understate true predictability in financial data. We also derive LLG-based refinements to the classic Hansen and Jagannathan (1991) bounds, analyze implications for parameter learning in general-equilibrium settings, and show that the LLG provides a natural mechanism for generating excess volatility.

Data attribution aims to explain model predictions by estimating how they would change if certain training points were removed, and is used in a wide range of applications, from interpretability and credit assignment to unlearning and privacy. Even in the relatively simple case of linear regressions, existing mathematical analyses of leading data attribution methods such as Influence Functions (IF) and single Newton Step (NS) remain limited in two key ways. First, they rely on global strong convexity assumptions which are often not satisfied in practice. Second, the resulting bounds scale very poorly with the number of parameters ($d$) and the number of samples removed ($k$). As a result, these analyses are not tight enough to answer fundamental questions such as "what is the asymptotic scaling of the errors of each method?" or "which of these methods is more accurate for a given dataset?" In this paper, we introduce a new analysis of the NS and IF data attribution methods for convex learning problems. To the best of our knowledge, this is the first analysis of these questions that does not assume global strong convexity and also the first explanation of [KATL19] and [RH25a]'s observation that NS data attribution is often more accurate than IF. We prove that for sufficiently well-behaved logistic regression, our bounds are asymptotically tight up to poly-logarithmic factors, yielding scaling laws for the errors in the average-case sample removals. \[ \mathbb{E}_{T \subseteq [n],\, |T| = k} \bigl[ \|\hatθ_T - \hatθ_T^{\mathrm{NS}}\|_2 \bigr] = \widetildeΘ\!\left(\frac{k d}{n^2}\right), \qquad \mathbb{E}_{T \subseteq [n],\, |T| = k} \bigl[ \|\hatθ_T^{\mathrm{NS}} - \hatθ_T^{\mathrm{IF}}\|_2 \bigr] = \widetildeΘ\!\left( \frac{(k + d)\sqrt{k d}}{n^2} \right). \]

We resolve a 30-year-old open problem concerning the power of unlabeled data in online learning by tightly quantifying the gap between transductive and standard online learning. In the standard setting, the optimal mistake bound is characterized by the Littlestone dimension $d$ of the concept class $H$ (Littlestone 1987). We prove that in the transductive setting, the mistake bound is at least $Ω(\sqrt{d})$. This constitutes an exponential improvement over previous lower bounds of $Ω(\log\log d)$, $Ω(\sqrt{\log d})$, and $Ω(\log d)$, due respectively to Ben-David, Kushilevitz, and Mansour (1995, 1997) and Hanneke, Moran, and Shafer (2023). We also show that this lower bound is tight: for every $d$, there exists a class of Littlestone dimension $d$ with transductive mistake bound $O(\sqrt{d})$. Our upper bound also improves upon the best known upper bound of $(2/3)d$ from Ben-David, Kushilevitz, and Mansour (1997). These results establish a quadratic gap between transductive and standard online learning, thereby highlighting the benefit of advance access to the unlabeled instance sequence. This contrasts with the PAC setting, where transductive and standard learning exhibit similar sample complexities.

Efforts to improve Kolmogorov-Arnold networks (KANs) with architectural enhancements have been stymied by the complexity those enhancements bring, undermining the interpretability that makes KANs attractive in the first place. Here we study overprovisioned architectures combined with sparsification to learn compact, interpretable KANs without sacrificing accuracy. Crucially, we focus on differentiable sparsification, turning architecture search into an end-to-end optimization problem. Across function approximation benchmarks, dynamical systems forecasting, and real-world prediction tasks, we demonstrate competitive or superior accuracy while discovering substantially smaller models. Overprovisioning and sparsification are synergistic, with the combination outperforming either alone. The result is a principled path toward models that are both more expressive and more interpretable, addressing a key tension in scientific machine learning.