# lagrange Published articles for lagrange. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## SIFR: Stable Signed Flow Reinforcement DevFeed: [SIFR: Stable Signed Flow Reinforcement]() Original publisher: [Read original article]() Author: Runlong Liao, Xingqian Xu, Lizhang Chen, Baiyu Su, Yufan Huang, Zhonghao Wang, Akash Gokul, Ming-Yu Liu, Qiang Liu, Yogesh Balaji Published: 2026-11-09T05:00:00Z Content type: article Language: en Sources: [arXiv: Machine Learning]() Topics: [Reinforcement learning](), [Fine-tuning](), [post-training]() Tags: [cosmos-3](), [cs-ai](), [cs-cv](), [cs-lg](), [fine-tuning](), [flow](), [flux](), [lagrange](), [rl](), [sd3-5](), [training]() ### AI overview SIFR aims to stabilize reward fine-tuning of flow models by combining signed advantages, which promote high-reward and suppress low-reward samples, with a constraint on velocity drift from a reference model. A Lagrange multiplier adapts regularization to a prescribed drift budget. The authors say existing flow RL methods implicitly minimize the signed loss and differ in their regularization choices. In evaluations across 12 backbone-reward pairs using preference and rule-based rewards, SIFR reportedly matched or exceeded existing methods, reached their peak rewards faster, and continued improving during long training where others plateaued or collapsed. Experiments covered SD3.5 Medium (2.5B), FLUX.2 Klein (9B), and Cosmos 3 Super Text2Image (64B). ### Source excerpt arXiv:2611.09175v1 Announce Type: new Abstract: Reward-based fine-tuning of flow models requires effectively promoting high-reward and suppressing low-reward samples, while maintaining training stability over many rounds of iterative updates. Iterative reward fine-tuning provides a simple starting point, but the positive reward weights give no explicit unfitting signal, and repeated reweighting concentrates the distribution and causes collapse. We propose Stable Signed Flow Reinforcement (SIFR), which replaces the reward weights by signed advantages to supply the missing unfitting signal, and constrains the velocity drift from a reference model to stabilize the updates, with a Lagrange multiplier that adapts the regularization strength to a prescribed drift budget. We further show that existing flow RL methods fall within this framework: they implicitly minimize the signed loss, differing in their regularization choices. We validate SIFR on SD3.5 Medium (2.5B) and FLUX.2 Klein (9B) and scale it to Cosmos 3 Super Text2Image (64B): across all 12 backbone--reward pairs with preference and rule-based rewards, it matches or exceeds existing methods, reaches their peak rewards much faster, and keeps improving over long training where they plateau or collapse. ## Emergence of the transformer architecture as the score of entropic laws of random measures DevFeed: [Emergence of the transformer architecture as the score of entropic laws of random measures]() Original publisher: [Read original article]() Author: Giacomo Greco Published: 2026-10-30T04:00:00Z Content type: article Language: en Sources: [arXiv: Artificial Intelligence]() Topics: [Transformer](), [model architecture]() Tags: [attention](), [axiom](), [choice](), [cs-ai](), [entropy](), [interaction](), [key](), [lagrange](), [language](), [law](), [masking](), [math-pr](), [matrix](), [model](), [next](), [one](), [order](), [parameter](), [random](), [score](), [stat-ml](), [transformer](), [transformer-architecture]() ### AI overview The paper derives a transformer head as the score of a last-token law in a maximum-entropy model of language. It treats sentences as empirical token measures and selects a law closest in relative entropy to a structure-free prior while matching an observed sentence statistic. Truncating Luce's choice axiom at the lowest relational order yields the softmax rule, with an attention cross-moment as its mean parameter. Query-key and value matrices arise as separate Lagrange multipliers for constraints within a sentence and across the language, respectively, so they are untied. For ordered sentences, a masked reference choice rule gives causal masking; using a next-token cross-moment makes the score equal to one causally masked attention head. The abstract also discusses multi-head attention, weight-tied unembedding, and a heuristic interpretation of residual blocks as projected Langevin steps. ### Source excerpt arXiv:2610.39614v1 Announce Type: new Abstract: We show that the transformer architecture emerges as the score of last-token laws in a natural statistical model of language, where sentences are empirical measures of tokens, a language is a law on $P(S^d)$, and our prior belief is that sentences carry no structure at all. By the maximum-entropy principle, we select the law closest to this prior in relative entropy among those reproducing an observed sentence-level statistic. We discuss the relevance of the statistic in the posterior interaction between tokens and we build a statistic, able to capture that interaction, with Luce's choice axiom truncated at the lowest relational order. This procedure gives the softmax choice rule, with the attention cross-moment as its mean parameter. The query--key matrix is then the Lagrange multiplier of a maximum-entropy problem inside each sentence, and the value matrix the multiplier of the constraint over the language, so that the two are untied. Under the optimal entropic law we then show that a transformer head appears in the score of the last-token law. For ordered sentences, causal masking follows from the masked reference choice rule, and with a next-token cross-moment the score equals one causally masked attention head. We also discuss multi-head attention, weight-tied unembedding, and a heuristic reading of residual blocks as projected Langevin steps. ## Local and Sparse Kernel Operator Learning for Incompressible Flows: An Operator Lagrange Formulation DevFeed: [Local and Sparse Kernel Operator Learning for Incompressible Flows: An Operator Lagrange Formulation]() Original publisher: [Read original article]() Author: Ramansh Sharma, Matthew Lowery, Houman Owhadi, Varun Shankar Published: 2026-10-22T04:00:00Z Content type: article Language: en Sources: [arXiv: Machine Learning]() Topics: [Learning](), [Neural Network]() Tags: [benchmarks](), [cs-lg](), [kernel](), [lagrange](), [learning](), [local](), [operator](), [operator-learning]() ### AI overview The method makes kernel operator learning for incompressible flows sparse while preserving divergence-free outputs. It replaces global output reconstruction with parallel local divergence-free interpolants and uses a compactly supported kernel for sparse regression; an optional affine input component captures global trends. Operator Lagrange evaluation requires one sparse solve per new input and avoids forming local coefficients. The abstract reports that this approach avoids solve-error amplification from cancellation among spatial coefficients. Across ten benchmarks, selected models were sparse nearly everywhere; affine augmentation reduced forced isotropic turbulence error by more than an order of magnitude. The sparse models were at least as accurate as the authors' earlier dense method and more accurate than neural operators. Lagrange evaluation matched direct evaluation with local reconstructions and was up to nearly three orders more accurate with global ones. ### Source excerpt arXiv:2610.29009v1 Announce Type: new Abstract: We present a local and sparse kernel method for learning the solution operators of incompressible flows. It builds on our earlier property-preserving kernel method, whose global output reconstruction and dense operator regression scale poorly. Our method replaces the global output reconstruction by local divergence-free interpolants with divergence-free polynomials of any degree, solved in parallel on the GPU. These interpolants are divergence free, periodic on periodic domains, and, with a multiscale kernel, approach Kolmogorov statistics under refinement. A compactly supported kernel, positive semidefinite in any input dimension, makes the operator regression sparse. Its support radius, and hence the sparsity, is fitted to data, and optional affine polynomials in the input capture global trends. Operator Lagrange weights join the two. They cost one sparse solve per new input and act on the training fields before the output reconstruction, so their local coefficients are never formed. We show that, unlike direct evaluation, Lagrange evaluation does not amplify solve errors by the cancellation among spatial coefficients. On ten fluid benchmarks, validation selects sparse matrices nearly everywhere, with densities that mostly decrease as the training set grows, and affine augmentation lowers the error by more than an order of magnitude for forced isotropic turbulence. These sparse models are at least as accurate as our earlier dense method and more accurate than neural operators. Lagrange evaluation is as accurate as direct evaluation with local output reconstructions, and up to nearly three orders of magnitude more accurate with global ones. We also present GPU timings and storage. ## A Reminder of Lagrange Multipliers for Optimization Problems DevFeed: [A Reminder of Lagrange Multipliers for Optimization Problems]() Original publisher: [Read original article]() Published: 2013-11-30T09:00:50Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Optimization](), [function](), [Variable]() Tags: [convex-functions](), [function](), [gradient-descent](), [lagrange](), [lagrange-multipliers](), [linear-algebra](), [optimization](), [problems](), [variable](), [vectors]() ### AI overview A tutorial-style reminder of Lagrange multipliers for optimization problems. It reviews gradients, partial derivatives, dot products, and directional change for multivariable functions. ### Source excerpt For a while I've been meaning to do some more advanced posts on optimization problems of all flavors. One technique that comes up over and over again is Lagrange multipliers, so this post is going to be a leisurely reminder of that technique. I often forget how to do these basic calculus-type things, so it's good practice. We will assume something about the reader's knowledge, but it's a short list: know how to operate with vectors and the dot product, know how to take a partial derivative, and know that in single-variable calculus the local maxima and minima of a differentiable function $ f(x)$ occur when the derivative $ f'(x)$ vanishes. ## Groups -- A Primer DevFeed: [Groups -- A Primer]() Original publisher: [Read original article]() Published: 2012-12-08T23:42:10Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [structure](), [Math and Logic](), [Cryptography]() Tags: [applications](), [group-actions](), [groups](), [lagrange](), [mathematics](), [public-key](), [set-theory](), [structure](), [symmetry](), [theory]() ### AI overview This primer introduces group theory as the study of algebraic structures that can describe operations beyond ordinary numbers. It motivates the subject through examples involving symmetry, squares, and Rubik's Cube, and notes applications including public-key cryptography, error detection, crystal structures, particle physics, and mathematical biology. ### Source excerpt The study of groups is often one's first foray into advanced mathematics. In the naivete of set theory one develops tools for describing basic objects, and through a first run at analysis one develops a certain dexterity for manipulating symbols and definitions. But it is not until the study of groups that one must step back and inspect the larger picture. The main point of that picture (and indeed the main point of a group) is that algebraic structure can be found in the most unalgebraic of settings.