Effective sub-typing (also known as grouping or clustering) of patients using their electronic health record (EHR) data can greatly inform precision medicine efforts. However, subtyping temporal EHR datasets is known to be challenging due to inherent EHR issues, including complexity and irregularity. In this study, we propose a self-supervised Mamba-based model that learns effective EHR representations and enables enhanced patient subtyping. We evaluate the proposed model on public and private real-world EHR datasets to classify the data based on the available labels and subtype patients based on the representations learned from the model. Through an extensive set of experiments, we demonstrate that our model's design choices lead to better performance compared to competitive baseline models for prediction. Moreover, we evaluate several clustering techniques to demonstrate that our findings offer valuable insights into subtyping patients based on temporal records from EHR models\footnote{Our implementations are available at https://github.com/healthylaife/triplet_mamba.

In this paper, we propose a novel learning paradigm, termed "Chain-of-Model" (CoM), which incorporates the causal relationship into the hidden states of each layer as a chain style, thereby introducing great scaling efficiency in model training and inference flexibility in deployment. We introduce the concept of "Chainof-Representation" (CoR), which formulates the hidden states at each layer as a combination of multiple sub-representations (i.e., chains) at the hidden dimension level. In each layer, each chain from the output representations can only view all of its preceding chains in the input representations. Consequently, the model built upon CoM framework can progressively scale up the model size by increasing the chains based on the previous models (i.e., chains), and offer multiple sub-models at varying sizes for elastic inference by using different chain numbers. Based on this principle, we devise Chain-of-Language-Model (CoLM), which incorporates the idea of CoM into each layer of Transformer architecture. Based on CoLM, we further introduce CoLM-Air by introducing a KV sharing mechanism, that computes all keys and values within the first chain and then shares across all chains. This design demonstrates additional extensibility, such as enabling seamless LM switching, prefilling acceleration and so on. Experimental results demonstrate our CoLM family can achieve comparable performance to the standard Transformer, while simultaneously enabling greater flexiblity, such as progressive scaling to improve training efficiency and offer multiple varying model sizes for elastic inference, paving a a new way toward building language models. Our code will be released in the future at: https://github.com/microsoft/CoLM.

A common approach for accelerating optimization algorithms is to minimize the loss achieved in a fixed time, which enables a differentiable framework with respect to the algorithm's hyperparameters. In contrast, the complementary objective of minimizing the time to reach a target loss is traditionally considered non-differentiable. To address this limitation, we propose a differentiable discrete stopping time and theoretically justify it based on its connection to continuous differential equations. We design an efficient algorithm to compute its sensitivities, thereby enabling a new differentiable formulation for directly accelerating algorithms. We demonstrate its effectiveness in applications such as online hyperparameter tuning and learning to optimize. Our proposed methods show superior performance in comprehensive experiments across various problems, which confirms their effectiveness.

Robust Subspace Recovery (RSR) aims to identify an underlying d-dimensional subspace from a dataset heavily corrupted by outliers. Complexity-theoretic results establish a threshold for the problem's computational hardness based on the dimensionscaled signal-to-noise ratio (DS-SNR): the problem is SSE-hard when the DS-SNR is strictly less than 1, and solvable via practical algorithms when it is greater than 1 under general position assumptions. However, the exact behavior of practical algorithms at the critical boundary DS-SNR = 1 has remained unknown. Specifically, we prove that TME converges exactly to the true subspace for DS-SNR 1 under a new stability condition, which is less restrictive than the general position assumptions used in prior literature. I. Introduction Robust Subspace Recovery (RSR) is a fundamental problem in robust statistics, machine learning, and computer vision. The primary goal of RSR is to identify an underlying low-dimensional linear subspace from a dataset that is heavily corrupted by outliers. The standard formulation of the noiseless RSR problem assumes a dataset X = {xi}Ni=1 RD consisting of n1 inliers lying exactly on a d-dimensional linear subspace L RD, and n0 outliers lying strictly off L . We refer to such a dataset as a noiseless inlier-outlier dataset, where the total number of points is N = n0 +n1. The central algorithmic question in noiseless RSR is under what conditions one can exactly and efficiently recover the underlying d-subspace L . A natural metric for characterizing the difficulty of this problem is the ratio of inliers to outliers, n1/n0, which can be viewed as a signal-to-noise ratio (SNR) [8], [11], [12]. This leads to the dimension-scaled SNR (DS-SNR), denoted by δS: δS:= n1/d n0/(D d) . Hardt and Moitra [5] established a fundamental lower bound, showing that when δS < 1, the noiseless RSR problem is Small Set Expansion (SSE)-hard, a property conjectured to be equivalent to NP-hardness [15]. In the special case of hyperplanes (d = D 1), they showed NP-hardness by invoking a result from [7]. The noiseless RSR problem is SSE-hard if δS < 1.

Topological Data Analysis (TDA) offers a principled, intrinsic lens for comparing neural representations. However, existing paired topological divergences (e.g., RTD) are limited by heuristic asymmetry and, more critically, unbounded scores that depend on sample size, hindering reliable cross-scenario benchmarking. To address these challenges, we develop a unified topological toolkit serving two complementary needs: fine-grained structural diagnosis and robust, standardized evaluation. First, we complete the RTD framework by introducing Symmetric Representation Topology Divergence (SRTD) and its efficient variant SRTD-lite. Beyond resolving the theoretical asymmetry of prior variants, SRTD consolidates diagnostic information into a single, comprehensive cross-barcode signature. This allows for precise localization of structural discrepancies and serves as an effective optimization objective without the overhead of dual directional computations. Second, to enable reliable benchmarking across heterogeneous settings, we propose Normalized Topological Similarity (NTS). By measuring the rank correlation of hierarchical merge orders, NTS yields a scale-invariant metric bounded between -1 and 1, effectively overcoming the scale and sample-dependence of unnormalized divergences. Experiments across synthetic and real-world deep learning settings demonstrate that our toolkit captures functional shifts in CNNs missed by geometric measures and robustly maps LLM genealogy even under distance saturation, offering a rigorous, topology-aware perspective that complements measures like CKA.

The minima of modern neural network loss functions are typically not isolated, rather they form connected components of reparameterization invariant solutions on the training data. Analytically characterizing these solutions is a hard problem, but sampling approaches are feasible. By construction, existing methods either spread over low-loss regions, and thus do not sample reparameterization invariant solutions exactly, or are inherently local, which limits exploration of other minima valleys. We propose sampling such reparameterization invariant models using a dynamical system based on kinetic energy, subject to a gravitational pull and a friction term that dissipates energy from the system. Our proposed sampler, DIMS, is guaranteed to sample exactly from the minimum level sets and depends on physically motivated hyperparameters which allows control over the exploration capabilities of the sampler. We consider uncertainty quantification in Bayesian inference as the motivating problem and observe improved performance compared to previously proposed approaches.

Organization: This supplementary material is presented in a format parallel to the main paper. The section numbers and titles are consistent with the main paper. But, here we also add one new section: Section 10 where we describe the societal impacts and possible negative impacts of the paper. Similarly, the Theorem numbers are consistent with the main paper, but we also have several additional theorems and lemmas which were not included in the main paper. GAN-type Objective for KLEstimation Let f be a discriminator, f: X IR. Let p(x) and q(x) be two probability density functions defined over the space X.

Swin transformer: Hierarchical vision transformer using shifted windows.