We introduce Triangular-Reference Schrödinger Bridges for Time Series (TR-SBTS), a conservative extension of the SBTS framework in which the Brownian reference is replaced by an intervalwise frozen, possibly degenerate diffusion reference, triangular across a hierarchy of latent volatility levels. The construction is a single entropy projection on the augmented state space, with the variational constraint imposed jointly across time and the latent levels and unfolded hierarchically by the disintegration of relative entropy. The variational core of SBTS is preserved: the entropy minimiser is the h-transform of the reference, and on each frozen interval the optimal dynamics admit a logarithmic-gradient drift formula on the affine leaves of the active covariance directions, valid even when the frozen covariance is rank-deficient. We establish stability of the frozen approximation and convergence of the corresponding regularised kernel estimators. The construction is realised through a finite-dimensional conditioning map assembled from three complementary reductions of the past -- a block PCR summary, a reference-aware Mahalanobis kernel on past increments induced by the runtime frozen covariance cumulants, and a past-window WLS drift regressor under the same reference metric -- together with a coupled state-covariance bridge step in which each latent level produces a dynamic reference for the level above, summarised by a covariance descriptor; the construction is evaluated on numerical experiments.

Obtaining stable diffusion-based samplers in high- and infinite-dimensional settings is challenging because errors can accumulate across high-frequency coordinates and make the dynamics unstable under refinement of the finite-dimensional approximation of the underlying function-space problem. Discretization is a typical source of such errors, and preconditioning with a suitable spectral decay is one way to control their accumulation. In this paper, we study this problem for preconditioned annealed Langevin dynamics (ALD) applied to Gaussian mixtures. We first show that Euler-Maruyama (EM) discretization, by treating the stiff linear part of the annealed score with a forward Euler step, imposes a stability constraint coupling the preconditioner with the annealed covariance scale. Together with the conditions ensuring dimension-uniform control of the annealed dynamics, this constraint forces the initial smoothed law to remain uniformly close to the target across dimensions. We then consider an exponential-integrator scheme that integrates the stiff linear part of the annealed score exactly. Under explicit spectral summability conditions coupling the smoothing covariance, the component covariance spectra, and the preconditioner, we prove a dimension-uniform Kullback-Leibler (KL) bound for this scheme. This bound can be made arbitrarily small, uniformly in dimension, by allowing enough time for annealing and then refining the time mesh accordingly. Importantly, these conditions allow regimes in which the KL divergence between the target and the initial smoothed law diverges with dimension, showing that the restrictions imposed by EM are scheme-dependent rather than intrinsic to ALD.

The signature is a canonical representation of a multidimensional path over an interval. However, it treats all historical information uniformly, offering no intrinsic mechanism for contextualising the relevance of the past. To address this, we introduce the Exponentially Weighted Signature (EWS), generalising the Exponentially Fading Memory (EFM) signature from diagonal to general bounded linear operators. These operators enable cross-channel coupling at the level of temporal weighting together with richer memory dynamics including oscillatory, growth, and regime-dependent behaviour, while preserving the algebraic strengths of the classical signature. We show that the EWS is the unique solution to a linear controlled differential equation on the tensor algebra, and that it generalises both state-space models and the Laplace and Fourier transforms of the path. The group-like structure of the EWS enables efficient computation and makes the framework amenable to gradient-based learning, with the full semigroup action parametrised by and learned through its generator. We use this framework to empirically demonstrate the expressivity gap between the EWS and both the signature and EFM on two SDE-based regression tasks.

In contextual dynamic pricing, a seller sequentially prices goods based on contextual information.

Recently, several fields used networked point processes to understand complex systems such as spiking biological neurons [36],social networks[8,42]geo-sensor networks[22],financial agents inmarkets[37],television records [48]and patient visits [11]. One ofthemain objectivesinthese analyses istouncoverthe causal relationships among the entities ofthe system, ortheinteraction structure among the nodes, which is also called thelatent network structure.

The framework also uses uncertainty measures on the Gaussian representations of thepreviously learned classes tofindthemost informativesamples tobelabeled in an increment. We evaluate our approach on the CORe-50 dataset and on a real humanoid robot for the object classification task.

In this paper, we study the finite-time convergence of nonlinear SA with multiplecoupledsequences.