Neural Legendre-Fenchel Transform with Hessian Preconditioning Improves Convex Conjugate Approximation
Researchers propose a new method using neural networks with Hessian-based preconditioning to compute Legendre-Fenchel transforms, which map functions to their convex conjugates. The approach leverages affine invariance to simplify the learning problem by transforming ill-conditioned functions into canonical forms. This advancement addresses computational challenges in convex analysis and machine learning, particularly for high-dimensional and ill-conditioned problems.
The Legendre-Fenchel transform is a fundamental operation in convex analysis and optimization that converts functions into their convex conjugates, but computing these conjugates becomes difficult when closed-form solutions are unavailable. Previous deep learning approaches using neural networks have struggled with ill-conditioned functions. This work reformulates the problem using projective polarity and introduces a Hessian-based preconditioning strategy that applies an affine deformation to transform the function into a canonical paraboloid form, whose conjugate is trivial to compute. A residual network then learns this simplified mapping with the original conjugate recovered through inverse deformation. The method requires modest computational overhead—one eigendecomposition at initialization and two matrix-vector multiplications per query. Experiments across diverse convex functions, including high-dimensional benchmarks, show improved convergence rates and numerical accuracy, with particularly strong gains for ill-conditioned problems.
What's missing
The paper acknowledges limitations and discusses scope of applicability but does not detail specific failure cases or problem classes where the method may not be suitable. Comparison with alternative preconditioning strategies beyond the proposed Hessian approach is not discussed.
What different sources said
- arXiv cs.LGCenter
Neural Legendre-Fenchel transform with Hessian Preconditioning
Related
Gut Bacteria Enzyme Found to Break Down Heat-Processed Food Compounds, Producing Novel Biogenic Amines
Researchers have discovered that an enzyme in common gut bacteria can degrade N-epsilon-carboxymethyllysine (CML), a compound formed during thermal food processing, producing previously unknown biogenic amines. The enzyme, ornithine decarboxylase SpeC from enterobacteria, acts on CML and related modified lysine derivatives through a low-level 'underground' catalytic activity. This finding suggests a previously unrecognized communication axis between thermally processed dietary compounds and gut microbial physiology, with potential implications for host health.
Full-Length Gene Sequencing Reveals Two Distinct Bacterial Communities in Black-Legged Ticks Expanding Into Canada
Researchers used Oxford Nanopore full-length 16S rRNA gene sequencing to characterize the microbiome of Ixodes scapularis black-legged ticks collected in Nova Scotia, Canada, distinguishing between tick-adapted bacteria and environmentally acquired bacteria. The study comes as I. scapularis — the primary vector of Lyme disease — is rapidly expanding northward into Canada due to climate change. The findings suggest that environmentally derived bacteria in tick microbiomes are not mere contamination, which has implications for how tick microbiome data is collected and interpreted across surveillance studies.
Study Identifies Metabolic Link Between Cell Envelope Stress and Biofilm Formation in Bacteria
Researchers have discovered that the metabolite acetyl-CoA directly inhibits enzymes that degrade the bacterial signaling molecule c-di-GMP, connecting cell envelope biosynthesis stress to biofilm formation in Pseudomonas aeruginosa. The study found that sub-inhibitory concentrations of antibiotics targeting early peptidoglycan biosynthesis — but not other antibiotic classes — elevate c-di-GMP levels by reducing phosphodiesterase activity, with acetyl-CoA competing for the enzyme active site. Because the relevant enzyme domain is broadly conserved across bacterial species, this checkpoint mechanism may be widespread and could have implications for understanding antibiotic-induced biofilm responses.