Causal representation learning for time series has developed strong identifiability results in discrete-time latent causal models, but identifiability in continuous-time latent stochastic differential equation (SDE) models remains largely open. We address this gap using environment-induced shifts in diffusion covariance. We study additive-noise latent SDEs observed through an unknown nonlinear diffeomorphism, with shared drift but environment-specific diffusion covariance. We show that two diagonal diffusion regimes with pairwise distinct coordinate-wise variance ratios identify the latent coordinates up to permutation and scaling, without any sparsity assumption on the drift. We first prove this result for linear Ornstein--Uhlenbeck systems and then extend it to general additive-noise latent SDEs. Under mild smoothness, the instantaneous drift-Jacobian causal graph is identifiable up to the same permutation. We propose a two-stage estimator for latent disentanglement and optional graph recovery; experiments on synthetic systems confirm the predicted identifiability boundary, and an application to Hardanger Bridge monitoring data illustrates the approach on real sensor trajectories.

We present LrcSSM, a non-linear recurrent model that processes long sequences as fast as today's linear state-space layers. By forcing its Jacobian matrix to be diagonal, the full sequence can be solved in parallel, giving O(TD) computational work and memory and only O(logT) sequential depth, for input-sequence length T and a state dimension D. Moreover, LrcSSM offers a formal gradient-stability guarantee that other input-varying systems such as Liquid-S4 and Mamba do not provide. Importantly, the diagonal Jacobian structure of our model results in no performance loss compared to the original model with dense Jacobian, and the approach can be generalized to other non-linear recurrent models, demonstrating broader applicability. On a suite of long-range forecasting tasks, we demonstrate that LrcSSM outperforms Transformers, LRU, S5, and Mamba.

The theoretical understanding of self-attention (SA) has been steadily progressing. A prominent line of work studies a class of SA layers that admit an energy function decreased by state updates. While it provides valuable insights into inherent biases in signal propagation, it often relies on idealized assumptions or additional constraints not necessarily present in standard SA. Thus, to broaden our understanding, this work aims to relax these energy constraints and provide an energy-agnostic characterization of inference dynamics by dynamical systems analysis. In more detail, we first consider relaxing the symmetry and single-head constraints traditionally required in energy-based formulations. Next, we show that analyzing the Jacobian matrix of the state is highly valuable when investigating more general SA architectures without necessarily admitting an energy function. It reveals that the normalization layer plays an essential role in suppressing the Lipschitzness of SA and the Jacobian's complex eigenvalues, which correspond to the oscillatory components of the dynamics. In addition, the Lyapunov exponents computed from the Jacobians demonstrate that the normalized dynamics lie close to a critical state, and this criticality serves as a strong indicator of high inference performance. Furthermore, the Jacobian perspective also enables us to develop regularization methods for training and a pseudo-energy for monitoring inference dynamics.

Polynomial Neural Networks (PNNs) possess a rich algebraic and geometric structure. However, their identifiability--a key property for ensuring interpretability-- remains poorly understood. In this work, we present a comprehensive analysis of the identifiability of deep PNNs, including architectures with and without bias terms. Our results reveal an intricate interplay between activation degrees and layer widths in achieving identifiability. As special cases, we show that architectures with non-increasing layer widths are generically identifiable under mild conditions, while encoder-decoder networks are identifiable when the decoder widths do not grow too rapidly compared to the activation degrees. Our proofs are constructive and center on a connection between deep PNNs and low-rank tensor decompositions, and Kruskal-type uniqueness theorems. We also settle an open conjecture on the dimension of PNN's neurovarieties, and provide new bounds on the activation degrees required for it to reach the expected dimension.

Many emerging applications--such as adversarial training, AI alignment, and robust optimization--can be framed as zero-sum games between neural nets, with von Neumann-Nash equilibria (NE) capturing the desirable system behavior. While such games often involve non-convex non-concave objectives, empirical evidence shows that simple gradient methods frequently converge, suggesting a hidden geometric structure. In this paper, we provide a theoretical framework that explains this phenomenon through the lens of hidden convexity and overparameterization. We identify sufficient conditions--spanning initialization, training dynamics, and network width--that guarantee global convergence to a NE in a broad class of non-convex min-max games. To our knowledge, this is the first such result for games that involve two-layer neural networks. Technically, our approach is twofold: (a) we derive a novel path-length bound for the alternating gradient descent-ascent scheme in min-max games; and (b) we show that the reduction from a hidden convex-concave geometry to two-sided Polyak-Łojasiewicz (PL) min-max condition hold with high probability under overparameterization, using tools from random matrix theory.

Large denoising diffusion models, such as Stable Diffusion, have been trained on billions of image-caption pairs to perform text-conditioned image generation. As a byproduct of this training, these models have acquired general knowledge about image statistics, which can be useful for other inference tasks. However, when confronted with sampling an image under new constraints, e.g.

We design a new unconstrained coordinate system where a $p\times p$ symmetric positive definite (SPD) matrix $Θ$ is represented by a reverse telescoping map $Θ(x)=\rm{RT}(x)$, with $x=(v,d,r)\in\mathbb{R}\times\mathbb{R}^{(p-1)}\times\mathbb{R}^{p(p-1)/2}$, representing respectively the log volume or log determinant; and the shape, as encoded by log relative diagonal scales and partial covariances among the nodes. This construction results in important properties not available in other charts, e.g., matrix logarithm, such as Jacobian depending on only the log-determinant. A useful feature of our construction is $x$ contains a lossless symbolic representation of both the matrix and its inverse. Many important computations involving a matrix and its inverse can be performed in $O(p^2)$ in the transformed domain, while it is the rendering of results in matrix forms (on demand) that must incur an $O(p^3)$ cost. Moreover, two unit-determinant matrices in the transformed domain can be joined by a straight line with pathwise unit determinant. For generative modeling, this allows designing a split volume-shape flow model trained by conditional flow matching for transporting the shape over the unit-determinant path, with a separate one-dimensional flow for transporting the volume or the determinant. The forbidding SPD constraint, tamed thus into a powerful guiding force, leads to the surprising insight that it is in some sense easier to design a volume-normalized shape flow for SPD compared to the unconstrained $\mathbb{R}^{p\times p}$, with no intrinsic notion of volume to aid normalization, unlike the determinant of SPD matrices. We apply our construction for up to $p=200$ in generative modeling of SPD matrices on a difficult synthetic bimodal target, and in generating brain connectivity networks by models trained on fMRI data; as well as in intrinsic diffusion on the SPD manifold.

Markov chain Monte Carlo (MCMC) methods are foundational algorithms for Bayesian inference and probabilistic modeling. However, most MCMC algorithms are inherently sequential and their time complexity scales linearly with the sequence length. Previous work on adapting MCMC to modern hardware has therefore focused on running many independent chains in parallel. Here, we take an alternative approach: we propose algorithms to evaluate MCMC samplers in parallel across the chain length. To do this, we build on recent methods for parallel evaluation of nonlinear recursions that formulate the state sequence as a solution to a fixed-point problem and solve for the fixed-point using a parallel form of Newton's method. We show how this approach can be used to parallelize Gibbs, Metropolis-adjusted Langevin, and Hamiltonian Monte Carlo sampling across the sequence length. In several examples, we demonstrate the simulation of up to hundreds of thousands of MCMC samples with only tens of parallel Newton iterations. Additionally, we develop two new parallel quasi-Newton methods to evaluate nonlinear recursions with lower memory costs and reduced runtime. We find that the proposed parallel algorithms accelerate MCMC sampling across multiple examples, in some cases by more than an order of magnitude compared to sequential evaluation.

Thus, SmoothDiff greatly enhances the usability (quality and speed) SmoothDiff's excellent speed and performance in a number of experiments and sible for shattered gradients and making it easy to implement. We demonstrate across a network architecture, directly targeting only the non4linearities respon4 leverages automatic differentiation to decompose the expected values of Jacobians yielding a speedup of over two orders of magnitude. Specifically, SmoothDiff work we propose a well founded novel method SmoothDiff to resolve this tradeoff demand, therefore in practice only few samples are used in SmoothGrad.

Latent component identification from unknown mixtures is a foundational challenge in machine learning, with applications in tasks such as self-supervised learning and causal representation learning. Prior work in (nICA) has shown that auxiliary signals---such as weak supervision---can support of conditionally independent latent components. More recent approaches explore structural assumptions, like sparsity in the Jacobian of the mixing function, to relax such requirements. In this work, we introduce (DICA), a framework that exploits the convex geometry of the mixing function's Jacobian. We propose a (J-VolMax) criterion, which enables latent component identification by encouraging diversity in their influence on the observed variables. Under suitable conditions, this approach achieves identifiability without relying on auxiliary information, latent component independence, or Jacobian sparsity assumptions. These results extend the scope of identifiability analysis and offer a complementary perspective to existing methods.