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.

We perform a thorough analysis of the formal and informal statements in the miniF2F benchmark from the perspective of an AI system that is tasked to participate in a math Olympiad consisting of the problems in miniF2F. In such setting, the model has to read and comprehend the problems in natural language, formalize them in Lean language, then proceed with proving the problems, and it will get credit for each problem if the formal proof corresponds to the original informal statement presented to the model. Our evaluation results reveal that the best accuracy of such pipeline can be about 36% using the SoTA models in the literature, considerably lower than the individual SoTA accuracies, 97% and 69% reported in the autoformalization and theorem proving literature. Analyzing the failure modes, we trace back a considerable portion of this drop to discrepancies between the formal and informal statements for more than half of the problems in miniF2F. We proceed with correcting all the errors, discrepancies and simplifications in formal and informal statements, and present the miniF2F-v2 with fully verified formal and informal statements and proofs. Evaluating the full theorem proving pipeline on miniF2F-v2 leads to the best accuracy of 70%, a significant improvement from the 40% on the original miniF2F, yet indicating considerable misalignment between the autoformalization models and theorem provers. Our deep analysis suggests that a higher quality benchmark can help the community better evaluate progress in the field of formal reasoning and also better diagnose the failure and success modes of autoformalization and theorem proving models.

Large Language Models (LLMs) have demonstrated remarkable performance in many applications, including challenging reasoning problems via chain-ofthought (CoT) techniques that generate "thinking tokens" before answering the questions. While existing theoretical works demonstrate that CoT with discrete tokens boosts the capability of LLMs, recent work on continuous CoT lacks a theoretical understanding of why it outperforms discrete counterparts in various reasoning tasks, such as directed graph reachability, a fundamental graph reasoning problem that includes many practical domain applications as special cases. In this paper, we prove that a two-layer transformer with D steps of continuous CoT can solve the directed graph reachability problem, where Dis the diameter of the graph, while the best known result of constant-depth transformers with discrete CoT requires O(n2) decoding steps where n is the number of vertices (D < n). In our construction, each continuous thought vector is a superposition state that encodes multiple search frontiers simultaneously (i.e., parallel breadth-first search (BFS)), while discrete CoT must choose a single path sampled from the superposition state, which leads to a sequential search that requires many more steps and may be trapped in local solutions. We also performed extensive experiments to verify that our theoretical construction aligns well with the empirical solution obtained via training dynamics. Notably, encoding of multiple search frontiers as a superposition state automatically emerges in training continuous CoT, without explicit supervision to guide the model to explore multiple paths simultaneously.

In this paper, we propose Jasmine, the first Stable Diffusion (SD)-based selfsupervised framework for monocular depth estimation, which effectively harnesses SD's visual priors to enhance the sharpness and generalization of unsupervised prediction. Previous SD-based methods are all supervised since adapting diffusion models for dense prediction requires high-precision supervision. In contrast, selfsupervised reprojection suffers from inherent challenges (e.g., occlusions, textureless regions, illumination variance), and the predictions exhibit blurs and artifacts that severely compromise SD's latent priors. To resolve this, we construct a novel surrogate task of mix-batch image reconstruction. Without any additional supervision, it preserves the detail priors of SD models by reconstructing the images themselves while preventing depth estimation from degradation. Furthermore, to address the inherent misalignment between SD's scale and shift invariant estimation and self-supervised scale-invariant depth estimation, we build the Scale-Shift GRU. It not only bridges this distribution gap but also isolates the fine-grained texture of SD output against the interference of reprojection loss. Extensive experiments demonstrate that Jasmine achieves SoTA performance on the KITTI benchmark and exhibits superior zero-shot generalization across multiple datasets. Project page and code are available at here.

Grid cells fire in triangular grid patterns, while place cells fire at specific locations and respond to contextual cues. How do these interacting systems support not only spatial encoding but also internally driven path planning, such as navigating to locations recalled from cues? Here, we propose a system-level theory of MEC-HC wiring that explains how grid and place cell patterns could be connected to enable cue-triggered goal retrieval, path planning, and reconstruction of sensory experience along planned routes. We suggest that place cells autoassociate sensory inputs with grid cell patterns, allowing sensory cues to trigger recall of goal-location grid patterns. We show analytically that grid-based planning permits shortcuts through unvisited locations and generalizes local transitions to long-range paths. During planning, intermediate grid states trigger place cell pattern completion, reconstructing sensory experiences along the route. Using a single-layer RNN modeling the HC-MEC loop with a planning subnetwork, we demonstrate these effects in both biologically grounded navigation simulations using RatatouGym and visually realistic navigation tasks using Habitat Sim. Codes for experiments, simulations, and vision encoder are available at 1,2,3.

Modern neural recording techniques such as two-photon imaging or Neuropixel probes allow to acquire vast time-series datasets with responses of hundreds or thousands of neurons. Contrastive learning is a powerful self-supervised framework for learning representations of complex datasets. Existing applications for neural time series rely on generic data augmentations and do not exploit the multi-trial data structure inherent in many neural datasets. Here we present TRACE, a new contrastive learning framework that averages across different subsets of trials to generate positive pairs. TRACE allows to directly learn a two-dimensional embedding, combining ideas from contrastive learning and neighbor embeddings. We show that TRACE outperforms other methods, resolving fine response differences in simulated data. Further, using in vivo recordings, we show that the representations learned by TRACE capture both biologically relevant continuous variation, cell-type-related cluster structure, and can assist data quality control.

Dexterous grasp generation is a fundamental challenge in robotics, requiring both grasp stability and adaptability across diverse objects and tasks. Analytical methods ensure stable grasps but are inefficient and lack task adaptability, while generative approaches improve efficiency and task integration but generalize poorly to unseen objects and tasks due to data limitations. In this paper, we propose a transfer-based framework for dexterous grasp generation, leveraging a conditional diffusion model to transfer high-quality grasps from shape templates to novel objects within the same category. Specifically, we reformulate the grasp transfer problem as the generation of an object contact map, incorporating object shape similarity and task specifications into the diffusion process. To handle complex shape variations, we introduce a dual mapping mechanism, capturing intricate geometric relationship between shape templates and novel objects.

As a new paradigm of visual content generation, autoregressive text-to-image models suffer from slow inference due to their sequential token-by-token decoding process, often requiring thousands of model forward passes to generate a single image. To address this inefficiency, we propose Speculative Jacobi-Denoising Decoding (SJD2), a framework that incorporates the denoising process into Jacobi iterations to enable parallel token generation in autoregressive models. Our method introduces a next-clean-token prediction paradigm that enables the pre-trained autoregressive models to accept noise-perturbed token embeddings and predict the next clean tokens through low-cost fine-tuning.

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Multi-bit quantization networks enable flexible deployment of deep neural networks by supporting multiple precision levels within a single model. However, existing approaches suffer from significant training overhead as full-dataset updates are repeated for each supported bit-width, resulting in a cost that scales linearly with the number of precisions. Additionally, extra fine-tuning stages are often required to support additional or intermediate precision options, further compounding the overall training burden. To address this issue, we propose two techniques that greatly reduce the training overhead without compromising model utility: (i) Weight bias correction enables shared batch normalization and eliminates the need for fine-tuning by neutralizing quantization-induced bias across bit-widths and aligning activation distributions; and (ii) Bit-wise coreset sampling strategy allows each child model to train on a compact, informative subset selected via gradient-based importance scores by exploiting the implicit knowledge transfer phenomenon. Experiments on CIFAR-10/100, TinyImageNet, and ImageNet-1K with both ResNet and ViT architectures demonstrate that our method achieves competitive or superior accuracy while reducing training time up to 7.88 . Our code is released at this link.