Estimating causal effects in industrial time series requires handling temporal dynamics, time-varying treatments, and unobserved confounders. Existing causal foundation models (CausalPFN, CausalFM) operate only on static cross-sectional data; neural temporal methods (CRN, G-Net) require per-dataset training; and concurrent temporal-PFN proposals have not been demonstrated at industrial scale. None output explicit per-pair reliability signals alongside their CATE estimates. We introduce Temporal Causal Prior-Data Fitted Networks (TCPFN), a foundation model for zero-shot temporal causal discovery with learned reliability signals. TCPFN makes four contributions: (1) a Causal Judgment Head that jointly predicts null-effect probability, confounding strength, identifiability, mediation fraction, and causal regime; (2) a mixed training prior covering six causal regimes (independent, direct, confounded, mediated, time-varying confounded, feedback) plus CausalFM-style front-door and instrumental-variable priors; (3) a discrete-token panel-data architecture with cross-attention masking that prevents inter-horizon leakage; (4) zero-shot inference at industrial scale via FAISS-based context selection and one-step posterior correction. On 19 benchmark datasets across five domains, TCPFN achieves competitive zero-shot causal discovery: AUROC 0.96 on Tennessee Eastman, 0.93 on SWaT, 0.98 on Causal Rivers, 0.97 on CAUSRCA. The null detector reaches NullF1 0.94, AUROC 0.99. TCPFN scales to V=1,275 on a proprietary Kraft pulp-and-paper dataset in 6 hours on a single GPU; PCMCI, a CPU-only library, on a V=666 sub-panel of the same data took 81.5 hours, extrapolating by O(V^2) to ~12.5 days at V=1,275. TCPFN's top edges identify cross-subsystem causal relationships while PCMCI's surface within-instrument controller-measurement coupling -- a scalability case study.

Inferring directed interactions between neural systems from EEG and MEG remains challenging due to noise, nonstationarity, and the high sample complexity of informationtheoretic estimators. Transfer Entropy (TE) provides a principled and model-free measure of directed information flow, however its practical estimation is not stable in finite data regimes (particularly as embedding dimension increases). This work introduces Embedded Polygon Symbolic Transfer Entropy (EPSTE), a framework that reframes TE estimation as a learnable problem operating on structured symbolic representations of local temporal morphology rather than raw signal amplitudes. Neural time series are decomposed into sequences of geometric primitives derived from local triplets of samples encoding complementary aspects of waveform structure such as magnitude, curvature and directional change. These primitives are discretised into symbolic tokens, yielding a compact but expressive state space over which symbolic TE is estimated. A recurrent neural network with attention-based multiple-instance learning is trained to predict surrogate-validated TE values from bags of symbolic temporal windows. The method is evaluated on source-reconstructed MEG data parcellated using the AAL90 atlas and compared against a standard symbolic baseline using identical architectures and supervision. The results demonstrate that while local window-level predictions are noisy, aggregation across trials and channel pairs yields stable directed dependencies. At the pair level, EPSTE achieves near-perfect recovery of ground-truth directed structure (Pearson r 0.99, R 0.98) and significantly lower absolute error than the baseline (Wilcoxon signed-rank test, p 2.9 10), indicating that representational geometry plays a critical role in enabling practical learnability of information-theoretic dependencies.

Current-status data arise when an event time is observed only through an indicator of whether it occurred before an examination time. This paper studies a nonparametric neural-network sieve maximum likelihood estimator of the conditional cumulative distribution function of the event time. Under Hölder smoothness assumptions, we establish an explicit convergence rate by combining approximation theory for rectified linear unit neural networks with empirical-process arguments. This result provides theoretical support for neural-network estimation and subsequent inference under current-status observation.

AdamW is the de facto optimizer for training large language models (LLMs), yet the theory behind it still lives mostly in finite-variance regimes. This is increasingly unsatisfying, as empirical evidence indicates that stochastic gradient noise in LLM pretraining is typically heavy-tailed. Recent work shows that sign-based optimizers such as Lion and Muon achieve sharp heavy-tailed rates, and that AdaGrad can also converge under heavy-tailed noise. However, no rigorous convergence theory for AdamW has yet been established in this regime. Can AdamW converge under the same heavy-tailed assumptions, or does its second-moment accumulator create a genuine obstruction? We formulate this as an open problem, prove a positive weighted-metric benchmark, and give a corridor lower-bound mechanism showing how denominator memory can hide large gradients.

Singular learning theory characterises the complexity of a deep network through the geometry of its loss singularities. The local learning coefficient (LLC), the standard estimator of Watanabe's real log canonical threshold (RLCT, $λ$), reads this geometry as an integrated Bayesian scalar through SGLD, which needs per-task calibration and $10^4$-$10^6$ forward-backward passes per checkpoint. We introduce Dead-Direction Signatures (DDS), a family of cheap closed-form spectral readings of singular structure: each reads a network's activation matrix or per-sample-gradient Fisher-Gram at a chosen layer, replacing the SGLD posterior chain with spectral linear algebra. The readings rest on a dead-direction framework that predicts a structural correlation between activation- and Fisher-side spectra at any singular minimum, and a rank-multiplicative volume identity that single-eigenvalue monitors cannot produce: the active-volume $\log\det^{+}(G)$ slope counts the dead directions, tracking the rank-deficit $r$ across $r \in \{1,2,3,4\}$ (slope ratios $2.0, 3.1, 4.0$ at $r{=}2,3,4$ against the predicted $2,3,4$), where the smallest eigenvalue is rank-blind. On reduced-rank regression with closed-form $λ$, calibrated LLC recovers $λ$ at $99\%$ mean and the DDS observables rank-track it at the framework-predicted sign; on a non-linear modular-addition transformer DDS separates $d_{\mathrm{model}}$ across eighteen orders of magnitude where calibrated LLC at the protocol budget is rank-flat. Complementary to LLC's integrated posterior reading, DDS gives a directional, layer-local handle on a network's dead directions, read in closed form from its activation and gradient spectra.

Many ranking-and-selection (R&S) problems arise in settings where information is noisy, structured, and expensive. In multi-fidelity experimentation, one can query cheap but biased proxies or expensive high-fidelity measurements; in dueling bandits, feedback arrives only through pairwise comparisons rather than direct rewards. These models are increasingly natural in engineering design, simulation optimization, preference learning for LLMs, and human-in-the-loop evaluation, where absolute scores are often unavailable or prohibitively costly and decisions must be made with a prescribed level of confidence. What makes these settings especially challenging is that the usual single-winner template is no longer sufficient. First, the answer map may be set-valued: in good-alternative or subset-selection problems, several answers can be simultaneously correct. Second, even when the true instance is answerable, a noisy estimate may temporarily fall outside the answerable set.

Clustering is a fundamental problem in statistics and machine learning. We propose the first one-bit clustering method for two-component sub-Gaussian mixture models. The method uses only one bit per entry of each sample obtained via a dithered quantizer. Under a mild non-spikiness condition on the cluster centers, we show that a variant of Lloyd's algorithm achieves a misclassification rate that decays exponentially with a signal-to-noise ratio comparable to that in the unquantized setting. This result further implies exact recovery under an explicit separation condition, which exceeds the optimal threshold for unquantized data by only a logarithmic factor. When the dimension $p$ is sufficiently large, the non-spikiness condition can be enforced by applying a random rotation using a Haar distributed matrix prior to quantization. In particular, it holds with high probability when $p \gtrsim 1$ for partial recovery and $p \gtrsim \log n \log\log n$ for exact recovery, where $n$ is the sample size. We also establish a minimax lower bound, showing that the misclassification rate and separation condition exhibit sharp constants in general. Numerical results are provided to corroborate the theory and demonstrate the efficacy of the proposed method.

Constraint-based causal discovery relies on repeated conditional independence tests, but fast nonparametric tests often sacrifice calibration, especially when variables depend on the conditioning set through nonlinear relationships. We introduce BLITZ (Broad-to-Local Independence Testing via residualiZation), a nonparametric conditional independence test designed to run well under a second while maintaining the accuracy needed for the thousands of queries performed by constraint-based causal discovery algorithms. BLITZ first removes broad smooth dependence on the conditioning set using low-order polynomial regression, then applies a small nonlinear feature map and residualizes those features with shallow tree regressions. The resulting statistic tests residual cross-covariance, with a moment-matched chi-square approximation to the null distribution. We show theoretically that the two-stage design reduces the effective complexity faced by the tree residualizers, allowing shallow trees to control residual conditional-mean bias while avoiding excessive overfitting. In simulations, BLITZ provides better null calibration than fast kernel, random-feature, and regression-based competitors while remaining among the fastest methods tested. In causal discovery experiments on synthetic graphs and flow-cytometry data, BLITZ yields more reliable endpoint orientations among retained adjacencies and competitive structural recovery. These results suggest that broad-to-local residualization is a practical route to calibrated, scalable nonparametric conditional independence testing for causal discovery.

Time series classification involves learning a mapping from a continuous, temporally ordered sequence of real-valued observations to a discrete response variable, like class labels. This task is fundamental in domains, including health monitoring, where the temporal structure of data is critical for accurate prediction. Dynamic Time Warping (DTW) is a standard technique for measuring similarity between sequences varying in time or speed. However, DTW is restricted to discrete point matching. To move beyond pairwise alignment, we propose a theoretical framework that learns mappings between real-valued functions. These mappings approximate the flow associated with the characteristic curves of a linear transport equation with a space-dependent velocity field, providing a diffeomorphic transformation between two time series. Using the method of characteristics, we transform this partial differential equation into ordinary differential equations (ODEs) modeling system dynamics. The objective function used to learn these ODEs derives from the fundamental theorem of calculus. To enable flexible, expressive representations of the velocity field, we utilize reproducing kernel Hilbert spaces and optimal control methods. Our method, Diffeomorphic Time Warping (DiffTW), provides a theoretically grounded dissimilarity measure. Using a 1-nearest neighbor classifier, DiffTW outperforms DTW on 60 of 86 datasets.

Learning instability is a long-standing problem across machine learning, but it is especially acute in the overparameterized regime that defines modern deep learning: large models fine-tuned or trained on limited data traverse flat loss landscapes with many nearly-equivalent minima, and stochastic factors (initialization, data order, dropout, hardware non-determinism) can route optimization to very different solutions. The rise of large pretrained models (LPMs) makes the problem more urgent: training cost is high, downstream data is often small, and repeated runs for variance reduction are prohibitive. We introduce \textbf{GRAIN} (\textbf{G}roup \textbf{A}ggregation via m\textbf{IN}-norm objective), a lightweight training algorithm that replaces the mean aggregation used in mini-batch optimization (both across mini-batches and within a mini-batch) with a min-norm convex combination of group-wise gradients. \mName guarantees a non-negative inner product between the aggregated update and every group gradient, resolving intra- and inner-batch gradient conflict, and retains an $\mathcal{O}(1/T)$ convergence rate comparable to SGD. Under mild smoothness and absolute-continuity assumptions, the min-norm solution differs almost surely from the arithmetic mean, which yields a uniform-stability bound for \mName strictly tighter than the standard bound for SGD. Empirically across generation, classification, and regression at LPM scale, \mName delivers consistent improvements in mean performance and reductions in run-to-run variance over a broad suite of tasks, with no extra training-time or storage cost beyond a single backward pass.