The recent paradigm shift to large-scale foundation models has brought about a new era for deep learning that, while has found great success in practice, has also been plagued by prohibitively expensive costs in terms of high memory consumption and compute. To mitigate these issues, there has been a concerted effort in post-hoc neural network pruning techniques that do not require costly retraining. Despite the considerable progress being made, existing methods often exhibit a steady drop in model performance as the compression increases. In this paper, we present a novel approach to compressing large transformers, coined OATS, that compresses the model weights by approximating each weight matrix as the sum of a sparse matrix and a low-rank matrix. Prior to the decomposition, the weights are first scaled by the second moment of their input embeddings, so as to ensure the preservation of outlier features recently observed in large transformer models. Without retraining, OATS achieves state-of-the-art performance when compressing large language models, such as Llama-3 and Phi-3, and vision transformers, such as Google's ViT and DINOv2, by up to $60\\%$, all while speeding up the model's inference on a CPU by up to $1.37\times$ compared to prior pruning methods.

Large language models (LLMs) often concentrate their attention on a few specific tokens referred to as *attention sinks*. Common examples include the first token, a prompt-independent sink, and punctuation tokens, which are prompt-dependent. While the tokens causing the sinks often lack direct semantic meaning, the presence of the sinks is critical for model performance, particularly under model compression and KV-caching. Despite their ubiquity, the function, semantic role, and origin of attention sinks—especially those beyond the first token—remain poorly understood. In this work, we conduct a comprehensive investigation demonstrating that attention sinks: *catch* a sequence of tokens, *tag* them using a common direction in embedding space, and *release* them back into the residual stream, where tokens are later retrieved based on the tags they have acquired. Probing experiments reveal these tags carry semantically meaningful information, such as the truth of a statement. These findings extend to reasoning models, where the mechanism spans more heads and explains greater variance in embeddings, or recent models with query-key normalization, where sinks remain just as prevalent. To encourage future theoretical analysis, we introduce a minimal problem which can be solved through the 'catch, tag, release' mechanism, and where it emerges through training.

Time-resolved single-cell omics data offers high-throughput, genome-wide measurements of cellular states, which are instrumental to reverse-engineer the processes underpinning cell fate. Such technologies are inherently destructive, allowing only cross-sectional measurements of the underlying stochastic dynamical system. Furthermore, cells may divide or die in addition to changing their molecular state. Collectively these present a major challenge to inferring realistic biophysical models. We present a novel approach, unbalanced probability flow inference, that addresses this challenge for biological processes modelled as stochastic dynamics with growth. By leveraging a Lagrangian formulation of the Fokker-Planck equation, our method accurately disentangles drift from intrinsic noise and growth. We showcase the applicability of our approach through evaluation on a range of simulated and real single-cell RNA-seq datasets. Comparing to several existing methods, we find our method achieves higher accuracy while enjoying a simple two-step training scheme.

We consider the Schrödinger bridge problem which, given ensemble measurements of the initial and final configurations of a stochastic dynamical system and some prior knowledge on the dynamics, aims to reconstruct the "most likely" evolution of the system compatible with the data. Most existing literature assume Brownian reference dynamics, and are implicitly limited to modelling systems driven by the gradient of a potential energy. We depart from this regime and consider reference processes described by a multivariate Ornstein-Uhlenbeck process with generic drift matrix $\mathbf{A} \in \mathbb{R}^{d \times d}$. When $\mathbf{A}$ is asymmetric, this corresponds to a non-equilibrium system in which non-gradient forces are at play: this is important for applications to biological systems, which naturally exist out-of-equilibrium. In the case of Gaussian marginals, we derive explicit expressions that characterise exactly the solution of both the static and dynamic Schrödinger bridge. For general marginals, we propose mvOU-OTFM, a simulation-free algorithm based on flow and score matching for learning an approximation to the Schrödinger bridge. In application to a range of problems based on synthetic and real single cell data, we demonstrate that mvOU-OTFM achieves higher accuracy compared to competing methods, whilst being significantly faster to train.

Pruning has emerged as a promising approach for compressing large-scale models, yet its effectiveness in recovering the sparsest of models has not yet been explored. We conducted an extensive series of 485,838 experiments, applying a range of state-of-the-art pruning algorithms to a synthetic dataset we created, named the Cubist Spiral. Our findings reveal a significant gap in performance compared to ideal sparse networks, which we identified through a novel combinatorial search algorithm. We attribute this performance gap to current pruning algorithms’ poor behaviour under overparameterization, their tendency to induce disconnected paths throughout the network, and their propensity to get stuck at suboptimal solutions, even when given the optimal width and initialization. This gap is concerning, given the simplicity of the network architectures and datasets used in our study. We hope that our research encourages further investigation into new pruning techniques that strive for true network sparsity.

Trajectory inference aims at recovering the dynamics of a population from snapshots of its temporal marginals. To solve this task, a min-entropy estimator relative to the Wiener measure in path space was introduced in [Lavenant et al., 2021], and shown to consistently recover the dynamics of a large class of drift-diffusion processes from the solution of an infinite dimensional convex optimization problem. In this paper, we introduce a grid-free algorithm to compute this estimator. Our method consists in a family of point clouds (one per snapshot) coupled via Schrödinger bridges which evolve with noisy gradient descent. We study the mean-field limit of the dynamics and prove its global convergence to the desired estimator. Overall, this leads to an inference method with end-to-end theoretical guarantees that solves an interpretable model for trajectory inference. We also present how to adapt the method to deal with mass variations, a useful extension when dealing with single cell RNA-sequencing data where cells can branch and die.

Understanding how cells change their identity and behaviour in living systems is an important question in many fields of biology. The problem of inferring cell trajectories from single-cell measurements has been a major topic in the single-cell analysis community, with different methods developed for equilibrium and non-equilibrium systems (e.g. haematopoeisis vs. embryonic development). We show that optimal transport analysis, a technique originally designed for analysing time-courses, may also be applied to infer cellular trajectories from a single snapshot of a population in equilibrium. Therefore, optimal transport provides a unified approach to inferring trajectories that is applicable to both stationary and non-stationary systems. Our method, StationaryOT, is mathematically motivated in a natural way from the hypothesis of a Waddington's epigenetic landscape. We implement StationaryOT as a software package and demonstrate its efficacy in applications to simulated data as well as single-cell data from Arabidopsis thaliana root development.