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Ivan Oseledets

Publications and source records attributed to Ivan Oseledets.

At least 19 recordsLinked to original sources
arXiv2026

Equidistribution of Solution Gradients in Physics-Informed Neural Networks via Coordinate Transformations

Physics-Informed Neural Networks (PINNs) often struggle to approximate solutions with sharp gradients due to optimization difficulties and spectral bias. Motivated by the principle of gradient equidistribution from the stretching approach in grid generation, which redistributes solution gradients to reduce functional complexity, we propose Coordinate Transformation PINNs (CTPINNs), which incorporate learnable coordinate transformations to map the physical domain into a computational domain where the physical solutions become approximately affine, making them easier for neural networks to learn. We propose two constructions of the transformation: an explicit construction based on Majorants and an implicit construction based on invertable neural networks. Experimentally, the explicit construction can improve accuracy for PDEs with available explicit coordinate transformations, and the implicit construction can be combined with other advanced PINN models as an efficient plug-in module to improve performance across a range of challenging PDE benchmarks.

math.NA↗
arXiv2026

Sigmoid Tensor Decomposition for Compact Representations of Binary Data

Nonlinear low-rank decompositions are receiving growing attention as a way to represent structured data with fewer parameters than conventional linear methods. Existing nonlinear approaches have focused mainly on matrices, which limits their ability to capture multidimensional structure. We propose a nonlinear tensor decomposition based on sigmoid nonlinearization in the Tucker and tensor-train formats. A binary tensor is represented by a low-rank real-valued score tensor, whose parameters are estimated by minimizing the binary cross-entropy loss. The resulting optimization problem is jointly nonconvex, but becomes convex with respect to one factor or score block when the remaining factors are fixed, as in the corresponding matrix case. We evaluate the method on several structured binary tensors and compare it with conventional Tucker and tensor-train decompositions across a range of common ranks. These results indicate that sigmoid nonlinearization is particularly effective in the low-rank regime, where it can provide more accurate binary reconstructions than conventional linear tensor decompositions at comparable model sizes.

math.NA↗
arXiv2026

Sanity Checks for Sparse Autoencoders: Do SAEs Beat Random Baselines?

Sparse Autoencoders (SAEs) have emerged as a promising tool for interpreting neural networks by decomposing their activations into sparse sets of human-interpretable features. Recent work has introduced multiple SAE variants and successfully scaled them to frontier models. Despite much excitement, a growing number of negative results in downstream tasks casts doubt on whether SAEs recover meaningful features. To directly investigate this, we perform two complementary evaluations. On a synthetic setup with known ground-truth features, we demonstrate that SAEs recover only $9\%$ of true features despite achieving $71\%$ explained variance, showing that they fail at their core task even when reconstruction is strong. To evaluate SAEs on real activations, we introduce three baselines that constrain SAE feature directions or their activation patterns to random values. Through extensive experiments across multiple SAE architectures, we show that our baselines match fully-trained SAEs in interpretability (0.87 vs 0.90), sparse probing (0.69 vs 0.72), and causal editing (0.73 vs 0.72). Together, these results suggest that SAEs in their current state do not reliably decompose models' internal mechanisms.

cs.LG↗
arXiv2026

VeriCypher: Named-Entity Verification for GraphRAG via Text-to-Cypher Generation

Knowledge graph question answering has traditionally focused on Resource Description Framework (RDF) graphs, SPARQL queries, and triple-based representations. Labeled Property Graphs (LPGs), however, pose a different challenge for LLM-based query generation: executable Cypher queries must align with schema structure, traversal direction, property keys, and literal entity values. This motivated us to introduce VeriCypher, an LLM-based text-to-Cypher framework for question answering over LPGs. It combines query generation conditioned on Cypher-like graph schema summaries with runtime execution, LLM-based query evaluation, and iterative correction where the central mechanism is named-entity verification: the system extracts node labels, relationship patterns, property keys, and literal values from generated queries, checks them against the target graph and schema, and turns mismatches into correction instructions for the next generation step. We evaluate VeriCypher on the CypherBench benchmark across multiple domains and LLM backbones. Compared with an LLM baseline, this method consistently improves answer correctness and query robustness. Ablation results further show that named-entity verification helps correct hallucinated schema elements and entity values, supporting more reliable LLM-based question answering over heterogeneous property graphs.

cs.AI↗
arXiv2026

TensorSpectra: Bridging the Spectral and Tensor Worlds for High-Dimensional PDEs

High-dimensional partial differential equations (PDEs) remain challenging for spectral methods due to the exponential growth of multidimensional coefficient spaces. We propose TensorSpectra, a low-rank spectral framework that directly parameterizes spectral coefficient tensors in canonical polyadic (CP) form, reducing representation complexity from $\mathcal{O}(N^d)$ to $\mathcal{O}(dNr)$. TensorSpectra combines structure-preserving spectral calculus, de-aliased nonlinear interactions, operator-aware residual weighting, and progressive low-rank corrections to enable efficient PDEs solving in compressed coefficient space. We further establish a theoretical framework connecting spectral truncation, low-rank approximation, and optimization errors. Extensive experiments across five PDE families and 24 benchmark configurations demonstrate State-Of-The-Art accuracy among five representative baselines, with TensorSpectra consistently achieving the lowest relative $L^2$ errors. More broadly, our findings offer a new perspective on high-dimensional PDEs solving, suggesting that exploiting intrinsic low-rank structure in spectral coefficient space provides a viable alternative to conventional physical-space approaches.

math.NA↗
arXiv2026

NAG-GS: A Semi-Implicit Stochastic Optimizer with Large-Step Stability

Stochastic first-order methods are widely used for large-scale optimization and machine learning, where their performance often depends on the learning rate. We study NAG-GS, a stochastic momentum method based on a semi-implicit Gauss--Seidel discretization of a Nesterov-type flow. For strongly convex quadratics, we determine the exact range of stable step sizes for fixed damping and the step size that minimizes the worst-case spectral radius. When the damping equals the strong-convexity parameter, the method recovers the accelerated asymptotic factor $(\sqrtκ-1)/(\sqrtκ+1)$, where $κ$ is the condition number. We also prove that the quadratic iteration converges linearly with dynamic damping when its limiting fixed-damping iteration is stable. For general strongly convex, possibly nonsmooth objectives, we establish an $O(1/k)$ bound on the expected objective gap of the last iterate, using a conditionally unbiased stochastic first-order oracle with bounded second moment. The result holds with dynamic damping from any $γ_0>0$. We show that NAG-GS can also be written as a Heavy-Ball iteration with explicitly determined step sizes and momentum coefficients. We test the stochastic result on nonsmooth hinge-loss classification problems, and further evaluate NAG-GS on quadratic problems, logistic regression, convolutional neural networks, RoBERTa, and Vision Transformers. The experiments show that NAG-GS can remain effective at learning rates where standard optimizers become unstable or lose accuracy.

math.OC↗
arXiv2026

Matcha: Multi-Stage Riemannian Flow Matching for Accurate and Physically Valid Molecular Docking

Predicting a protein--ligand binding pose is central to structure-based drug design, yet current docking methods still struggle to combine speed, accuracy and physical plausibility. We introduce Matcha, which refines a ligand pose by three flow-matching stages in translation, rotation and torsion space, then minimizes, filters and ranks candidates with GNINA. Across the benchmarks considered, Matcha attains the strongest physical plausibility and runs about $28\times$ faster than modern large-scale co-folding models. Code and benchmarking artifacts are available at https://github.com/LigandPro/Matcha and archived at https://doi.org/10.5281/zenodo.23568093. Model checkpoints are archived at https://doi.org/10.5281/zenodo.23569055.

cs.LG↗
arXiv2026

Separating Thinking from Spelling for Efficient Generative Recommendation

Generative recommenders retrieve items by writing their semantic IDs (SIDs) code by code under trie-constrained beam search, an approach now deployed on large web platforms. Standard autoregressive interfaces pay for the SID length K twice: a history of T items enters the backbone as TK code tokens, and every beam-search step runs through the backbone, so returning B candidates costs O(BK) backbone decode tokens per user. The second cost often dominates, and it grows with the backbone, the SID depth and the candidate budget. We introduce Deco (decoding from concepts), which separates reading the history from writing the identifier. The backbone reads each history item as one fused token and condenses the history once into a compact next-item concept; a light KV-cached decoder conditioned on the concept and the generated prefix runs the entire beam search, while the identifiers, the trie and the autoregressive factorization stay unchanged. The decoder approaches full-backbone accuracy with cross-entropy alone, and distillation improves it further. On seven datasets from e-commerce, movies and music streaming, Deco achieves comparable or better accuracy than both a standard generative recommender and the full-backbone teacher with fused inputs, while delivering 3.2-8.6$\times$ their throughput under a 50 ms p95 latency budget in a production-style serving pipeline. Delegating beam search makes larger candidate pools and models practical: Deco generates 1000 candidates faster than its teacher generates 100, and with a 1.5B-parameter LLM backbone, it matches the 69M-parameter teacher's throughput while achieving higher accuracy.

cs.IR↗
arXiv2026

From Stories to Features: Scaling Concept Vectors for AI Alignment & Interpretability

We study feature vectors: directions in a language model's activation space that correspond to human-interpretable concepts, such as honesty or deception. Existing unsupervised methods like sparse autoencoders attempt to recover the whole feature space, often at the cost of feature quality and interpretability. We take a more pragmatic route: focusing on concepts relevant to AI safety and alignment, we generate synthetic concept-antagonist story pairs and use the difference of their mean model activations as a feature vector. We scale this approach to 1,036 alignment-relevant concepts using 2M synthetic stories across multiple languages and genres, which we publicly release. Across six models from three model families, our feature vectors reliably generalize to detecting their corresponding concepts in chat dialogues and web text. They form an interpretable geometry shared across models and can causally steer model behavior when injected into activations. We further use these features to analyze model representations during jailbreak attacks, mathematical and agentic reasoning. Our targeted feature construction thus offers a scalable route to studying concepts in language models, potentially extending to millions of features across arbitrary domains.

cs.CL↗
arXiv2026

What Limits Recursive Reasoning Models: Optimization, Architecture and Test-Time Scaling

Recursive reasoning models apply a small shared Transformer block many times to refine a latent state. This gives them large effective depth with few parameters and makes them strong on algorithmic tasks. Such compact solvers are natural candidates for tools that an LLM can call on narrow algorithmic subproblems. However, existing models such as HRM, TRM and URM differ in architecture, gradient propagation and training procedure simultaneously. This makes it hard to tell what drives their performance, and their optimization is still poorly understood and often unstable. In this work we address both of these gaps. First, we study these questions under a unified experimental pipeline spanning six algorithmic domains. Individual controlled ablations are performed on representative domains, while the resulting recipe is evaluated across the full suite. The study reveals a surprisingly simple recipe for stable and generalizable recursive reasoning: an intermediate gradient horizon, large physical batches and controlled updates of the recurrent state. An explicit hierarchical architecture is not needed. Second, we combine these findings into a stable 13.6M-parameter model that achieves the strongest overall performance among the evaluated recursive baselines, with particularly large gains on out-of-distribution generalization. It raises Arithmetic OOD accuracy to 71.2%, from 36.2% for the strongest baseline, while reaching 98.41% on Sudoku and 59.5% pass@2 on ARC-AGI-1. Our results show that, within the recursive architectures studied here, performance depends strongly on how recurrence is optimized and stabilized. More broadly, it shows how AI systems can be improved by optimizing their components one at a time.

cs.LG↗
arXiv2026

E-MoE: Enhanced Mixture-of-Experts for Non-Factorized Diffusion Language Models

Masked diffusion models (MDMs) generate sequences by progressively unmasking several tokens per denoising step, but their reverse process is typically factorized over positions, limiting sample quality in the few-step regime where diffusion's speed advantage over autoregressive decoding matters most. A recent line of work introduces a continuous Gaussian latent, trained as a variational autoencoder, to capture correlations across positions, but such approaches are prone to posterior collapse, where the latent is silently ignored. We propose Enhanced Mixture-of-Experts (E-MoE), which builds the reverse process as a mixture of factorized distributions over a discrete shared latent given by the expert-routing decisions of a Mixture-of-Experts (MoE) backbone, without increasing active parameters over the factorized baseline. Across synthetic multi-modal benchmarks, binarized MNIST, and LM1B, E-MoE improves few-step generation over factorized baselines.

cs.CL↗
arXiv2026

Evaluating Losslessness in Speculative Decoding Under Finite-Precision Inference

Lossless speculative decoding is typically defined at the algorithmic level: a speculative procedure proposes multiple tokens and a verification procedure is designed to preserve the output trajectory of an autoregressive reference model exactly. In practical neural inference, however, this guarantee is implemented using finite-precision floating-point computations, and discrete token selection can amplify small numerical differences into divergent generation trajectories. We investigate this distinction using Orthrus, a hybrid autoregressive-diffusion architecture that performs self-drafting and self-verification within a frozen autoregressive backbone, as a representative case study. Across 1,190 prompts from 12 domains, exact trajectory matching under BF16 occurs for only 45\% of the authors' checkpoint generations and 43% of those from our independently trained model. The probability of matching is strongly associated with the response-conditional perplexity of the autoregressive reference, indicating that trajectory divergence is not uniform across inputs. Despite these divergences, Orthrus does not exhibit systematic degradation on the evaluated downstream tasks. In contrast, FP32 inference yields exact trajectory matching on all evaluated prompts. These results demonstrate a gap between algorithmic losslessness and its implementation under finite-precision arithmetic, and motivate evaluating lossless speculative decoding at the level of exact generation trajectories as well as downstream task performance.

cs.CL↗
arXiv2026

Reasoning on the Simplex: Geometric Fixed-Point Models

Looped reasoners spend test-time compute by iterating a weight-tied map, but a small residual does not mean the state is a fixed point when that map lives in unconstrained latent space. We propose Geometric Fixed-Point Reasoning (GFPR), in which the iterated state is the prediction itself: a field of categorical beliefs on a product of simplices, whose argmax is the answer at every step. Because the state is a belief, task structure can be imposed through compact convex relaxations, either as structured readouts or directly in the recurrent state; in the latter case the update remains a continuous self-map, so a fixed point exists for any parameters. At about 7M parameters, GFPR reaches 95.1% exact match on Sudoku-Extreme, 92.0% on Maze-Hard, and 100% sequence accuracy on S_5 length 128, above the published FPRM numbers at the same scale. The same update also trains a 201M language model on FineWeb-Edu in which each site is a distribution over the vocabulary; with 24 Picard steps it is above GPT-2 small on four zero-shot multiple-choice tasks and above GPT-2 medium on ARC-Easy.

cs.AI↗
arXiv2026

LANTERN: Illuminating Hidden Mathematical Knowledge in Language Models

Language models can now prove theorems, but people still decide which problems to pursue. We ask whether a model's internal representations can help identify promising mathematical connections. We develop LANTERN, a fast, cost-efficient pipeline that uses a classifier over pretrained-model activations to rank candidate relations, followed by staged filtering, hypothesis generation, executable verification, and analytical checking. Applied to the On-Line Encyclopedia of Integer Sequences (OEIS), LANTERN ranked 50 million pairs among 10,000 frequently referenced sequences and produced 62 verified relations between pairs without an existing OEIS cross-reference. A content screen retained 13 relations worth presenting; nine of these are informative or insightful, including four which are entirely novel to the best of our knowledge: none appears in the OEIS or in our targeted literature search. The entire end-to-end process including classifier training, candidate ranking, filtering and verification took under 8 hours.

cs.CL↗
arXiv2026

Change the Product, Keep the Parameters: Associative Algebra Layers for Transformers

Fast matrix multiplication algorithms keep the product fixed and search for a cheaper way to evaluate it. We instead ask whether a Transformer's learned projections can use a different, cheaper product altogether. Building on an associative-algebra construction that replaces ordinary matrix multiplication with a sparser interaction table over the same weight blocks, we construct a family with quadratic arithmetic in the matrix dimension when the physical block size remains fixed, and derive finite-shape constraints for GPU execution. The construction is provably optimal for its bilinear rank by the Alder--Strassen bound and can be realized as row-typed rectangular projections compatible with causal masking and KV-cached decoding. We provide an empirical test of this approach by training two approximately 110M-parameter decoder-only Transformer LMs from the same recipe and 12.3B-token budget, differing only in their feed-forward layer: one uses ordinary dense matrix multiplication and the other uses the associative-algebra product. Across four prompt domains, the algebraic model achieves a 6.2--7.8\% increase in end-to-end generation throughput, while obtaining lower scores on all three reported downstream metrics. We treat these results as a feasibility and trainability check for the proposed approach at small scale, leaving further investigation to future work.

cs.LG↗
arXiv2026

Your Transformer Can Hold Two Thoughts at Once: Evidence of Linear Superposition in LLMs

While Large Language Models (LLMs) rely on highly non-linear components, in this work we demonstrate that they exhibit fundamental linearity: when inputs from distinct text streams are linearly combined, the model outputs a superposition of the individual next-token distributions. We term this the \textit{Superposition Linearity Hypothesis}. We provide evidence that superposition is an intrinsic property of the Transformer architecture rather than an emergent consequence of training; in fact, we observe that it tends to diminish as pretraining progresses. However, we demonstrate that linearity can be substantially restored through lightweight fine-tuning, significantly reducing the divergence between the predicted next-token distribution and the average of the individual next-token distributions. Finally, we introduce a guided decoding procedure that disentangles superposed outputs, enabling the simultaneous generation of two coherent continuations from a single forward pass.

cs.CL↗
arXiv2026

CoMa: Contextual Massing Generation with Vision-Language Models

Context-aware building massing is an important early-stage design task: given a site for buildings, a generated massing should not only fit the target parcel, but also relate to the scale, density, and morphology of its surrounding urban fabric. This task is naturally multimodal, since the target output should remain structured and editable, while the surrounding context, including other buildings or roads, can be represented as vector geometry, map imagery, or three-dimensional views. In this paper, we study contextual massing generation using vision-language models (VLMs) and analyze their performance on this task across different context modalities during training and inference. We assemble an experimental dataset of 12,845 Melbourne massings with parcel contours, structured 3D geometry, neighboring buildings, top-down views, and multi-view 3D context images. We also introduce a learned contextual relevance metric for evaluating whether generated massings are morphologically compatible with their surrounding context. Using Qwen3-VL models, we compare no-context, unimodal-context, and multimodal-context training regimes and evaluate inference performance under controlled combinations of modalities and amounts of context. The results show that model size strongly affects generation quality, multimodal training improves the use of individual modalities, and multimodal inference provides a stronger contextual signal than isolated context inputs.

cs.CV↗
arXiv2026

LLM-Microscope: Uncovering the Hidden Role of Punctuation in Context Memory of Transformers

We introduce methods to quantify how Large Language Models (LLMs) encode and store contextual information, revealing that tokens often seen as minor (e.g., determiners, punctuation) carry surprisingly high context. Notably, removing these tokens -- especially stopwords, articles, and commas -- consistently degrades performance on MMLU and BABILong-4k, even if removing only irrelevant tokens. Our analysis also shows a strong correlation between contextualization and linearity, where linearity measures how closely the transformation from one layer's embeddings to the next can be approximated by a single linear mapping. These findings underscore the hidden importance of filler tokens in maintaining context. For further exploration, we present LLM-Microscope, an open-source toolkit that assesses token-level nonlinearity, evaluates contextual memory, visualizes intermediate layer contributions (via an adapted Logit Lens), and measures the intrinsic dimensionality of representations. This toolkit illuminates how seemingly trivial tokens can be critical for long-range understanding.

cs.CL↗