Daily Papers

Standard simulations of Turing machines suggest a linear relationship between the temporal duration t of a run and the amount of information that must be stored by known simulations to certify, verify, or regenerate the configuration at time t. For deterministic multitape Turing machines over a fixed finite alphabet, this apparent linear dependence is not intrinsic: any length-t run can be simulated using O(t) work-tape cells via a Height Compression Theorem for succinct computation trees together with an Algebraic Replay Engine. In this paper we recast that construction in geometric and information-theoretic language. We interpret the execution trace as a spacetime DAG of local update events and exhibit a family of recursively defined holographic boundary summaries such that, along the square-root-space simulation, the total description length of all boundary data stored at any time is O(t). Using Kolmogorov complexity, we prove that every internal configuration has constant conditional description complexity given the appropriate boundary summary and time index, establishing that the spacetime bulk carries no additional algorithmic information beyond its boundary. We express this as a one-dimensional computational area law: there exists a simulation in which the information capacity of the active "holographic screen'' needed to generate a spacetime region of volume proportional to t is bounded by O(t). In this precise sense, deterministic computation on a one-dimensional work tape admits a holographic representation, with the bulk history algebraically determined by data residing on a lower-dimensional boundary screen.

Given a finite poset mathcal P, the hypercube-height, denoted by h^*(mathcal P), is defined to be the largest h such that, for any natural number n, the subsets of [n] of size less than h do not contain an induced copy of mathcal P. The hypercube-width, denoted by w^*(mathcal P), is the smallest w such that the subsets of [w] of size at most h^*(mathcal P) contain an induced copy of mathcal P. In other words, h^*(mathcal P) asks how `low' can a poset be embedded, and w^*(mathcal P) asks for the first hypercube in which such an `optimal' embedding occurs. These notions were introduced by Bastide, Groenland, Ivan and Johnston in connection to upper bounds for the poset saturation numbers. While it is not hard to see that h^*(mathcal P)leq |mathcal P|-1 (and this bound can be tight), the hypercube-width has proved to be much more elusive. It was shown by the authors mentioned above that w^*(mathcal P)leq|mathcal P|^2/4, but they conjectured that in fact w^*(mathcal P)leq |mathcal P| for any finite poset mathcal P. In this paper we prove this conjecture. The proof uses Hall's theorem for bipartite graphs as a precision tool for modifing an existing copy of our poset.

We present a novel global compression framework for deep neural networks that automatically analyzes each layer to identify the optimal per-layer compression ratio, while simultaneously achieving the desired overall compression. Our algorithm hinges on the idea of compressing each convolutional (or fully-connected) layer by slicing its channels into multiple groups and decomposing each group via low-rank decomposition. At the core of our algorithm is the derivation of layer-wise error bounds from the Eckart Young Mirsky theorem. We then leverage these bounds to frame the compression problem as an optimization problem where we wish to minimize the maximum compression error across layers and propose an efficient algorithm towards a solution. Our experiments indicate that our method outperforms existing low-rank compression approaches across a wide range of networks and data sets. We believe that our results open up new avenues for future research into the global performance-size trade-offs of modern neural networks. Our code is available at https://github.com/lucaslie/torchprune.

The Shadowing Principle of Manin has proved a valuable tool for addressing questions of quantitative topology raised by Gromov in the late 1900s. The principle informally provides a way for bounded algebraic maps between differential graded algebras to be translated into nearby genuine maps between their geometric realizations. We extend this principle to finite towers of principal K(G,n) fibrations, and in particular apply this construction to nilpotent spaces. As a specific application of the extended principle, we provide upper bounds on the asymptotic behavior of volumes of nullhomotopies of Lipschitz maps into nilpotent spaces. We further refine these bounds in the case when c = 1 to nearly meet those of the simply connected setting. We similarly refine these bounds in the event the target space is coformal, and demonstrate that the bounds in this setting are nearly sharp.

We give an alternative proof of the Schoen--Simon--Yau curvature estimates and associated Bernstein-type theorems (1975), and extend the original result by including the case of 6-dimensional (stable minimal) immersions. The key step is an ε-regularity theorem, that assumes smallness of the scale-invariant L^2 norm of the second fundamental form. Further, we obtain a graph description, in the Lipschitz multi-valued sense, for any stable minimal immersion of dimension ngeq 2, that may have a singular set Σ of locally finite H^{n-2}-measure, and that is weakly close to a hyperplane. (In fact, if H^{n-2}(Σ)=0, the conclusion is strengthened to a union of smooth graphs.) This follows directly from an ε-regularity theorem, that assumes smallness of the scale-invariant L^2 tilt-excess (verified when the hypersurface is weakly close to a hyperplane). Specialising the multi-valued decomposition to the case of embeddings, we recover the Schoen--Simon theorem (1981). In both ε-regularity theorems the relevant quantity (respectively, length of the second fundamental form and tilt function) solves a non-linear PDE on the immersed minimal hypersurface. The proof is carried out intrinsically (without linearising the PDE) by implementing an iteration method à la De Giorgi (from the linear De Giorgi--Nash--Moser theory). Stability implies estimates (intrinsic weak Caccioppoli inequalities) that make the iteration effective despite the non-linear framework. (In both ε-regularity theorems the method gives explicit constants that quantify the required smallness.)

We consider distributed image transmission over a noisy multiple access channel (MAC) using deep joint source-channel coding (DeepJSCC). It is known that Shannon's separation theorem holds when transmitting independent sources over a MAC in the asymptotic infinite block length regime. However, we are interested in the practical finite block length regime, in which case separate source and channel coding is known to be suboptimal. We introduce a novel joint image compression and transmission scheme, where the devices send their compressed image representations in a non-orthogonal manner. While non-orthogonal multiple access (NOMA) is known to achieve the capacity region, to the best of our knowledge, non-orthogonal joint source channel coding (JSCC) scheme for practical systems has not been studied before. Through extensive experiments, we show significant improvements in terms of the quality of the reconstructed images compared to orthogonal transmission employing current DeepJSCC approaches particularly for low bandwidth ratios. We publicly share source code to facilitate further research and reproducibility.

For every integer Nge4, we construct a fixed smooth, non-locally-conformally-flat metric on the n-dimensional unit sphere S^n for which the constant Q_{2N}-curvature equation admits an L^infty-unbounded sequence of positive solutions. The construction applies for nge2N+20 when N=4,5, nge2N+19 when 6le Nle8, nge2N+18 when 9le Nle17, and nge2N+17 when Nge18. For each L in N cup {0}, the metric can be chosen arbitrarily close to the round metric in the C^L norm. Together with the known cases N=1,2,3, this establishes noncompactness at every order, with dimension bounds that we expect to be optimal. We derive an explicit formula for the fixed-volume Hessian of total Q_{2N}-curvature in transverse-traceless (TT) directions at closed Einstein metrics satisfying Ric=(n-1)g. Building on Juhl's formulas, we identify this Hessian, for every NinN and n>2N, as a degree-N polynomial in the Lichnerowicz Laplacian. For the metric perturbations generated by algebraic Weyl tensors in the construction, the quadratic reduced energy with fixed bubble center is a positive multiple of this Hessian restricted to a four-dimensional TT space. Differentiation with respect to the bubble scale then yields a finite matrix whose exact sign analysis establishes the required indefiniteness in the stated dimension ranges. After continuation to real n, this matrix changes from negative definite to indefinite at n_N=2N+a_*+c_*N^{-1}+O(N^{-2}) as Ntoinfty, with a_*approx16.201871 and c_*approx13.512880, explaining the eventual bound nge2N+17.

The task of vision-based 3D occupancy prediction aims to reconstruct 3D geometry and estimate its semantic classes from 2D color images, where the 2D-to-3D view transformation is an indispensable step. Most previous methods conduct forward projection, such as BEVPooling and VoxelPooling, both of which map the 2D image features into 3D grids. However, the current grid representing features within a certain height range usually introduces many confusing features that belong to other height ranges. To address this challenge, we present Deep Height Decoupling (DHD), a novel framework that incorporates explicit height prior to filter out the confusing features. Specifically, DHD first predicts height maps via explicit supervision. Based on the height distribution statistics, DHD designs Mask Guided Height Sampling (MGHS) to adaptively decouple the height map into multiple binary masks. MGHS projects the 2D image features into multiple subspaces, where each grid contains features within reasonable height ranges. Finally, a Synergistic Feature Aggregation (SFA) module is deployed to enhance the feature representation through channel and spatial affinities, enabling further occupancy refinement. On the popular Occ3D-nuScenes benchmark, our method achieves state-of-the-art performance even with minimal input frames. Source code is released at https://github.com/yanzq95/DHD.

Scaling laws have shaped recent advances in machine learning by enabling predictable scaling of model performance based on model size, computation, and data volume. Concurrently, the rise in computational cost for AI has motivated model compression techniques, notably quantization and sparsification, which have emerged to mitigate the steep computational demands associated with large-scale training and inference. This paper investigates the interplay between scaling laws and compression formats, exploring whether a unified scaling framework can accurately predict model performance when training occurs over various compressed representations, such as sparse, scalar-quantized, sparse-quantized or even vector-quantized formats. Our key contributions include validating a general scaling law formulation and showing that it is applicable both individually but also composably across compression types. Based on this, our main finding is demonstrating both theoretically and empirically that there exists a simple "capacity" metric -- based on the representation's ability to fit random Gaussian data -- which can robustly predict parameter efficiency across multiple compressed representations. On the practical side, we extend our formulation to directly compare the accuracy potential of different compressed formats, and to derive better algorithms for training over sparse-quantized formats.

When training large-scale models, the performance typically scales with the number of parameters and the dataset size according to a slow power law. A fundamental theoretical and practical question is whether comparable performance can be achieved with significantly smaller models and substantially less data. In this work, we provide a positive and constructive answer. We prove that a generic permutation-invariant function of d objects can be asymptotically compressed into a function of polylog d objects with vanishing error, which is proved to be the optimal compression rate. This theorem yields two key implications: (Ia) a large neural network can be compressed to polylogarithmic width while preserving its learning dynamics; (Ib) a large dataset can be compressed to polylogarithmic size while leaving the loss landscape of the corresponding model unchanged. Implication (Ia) directly establishes a proof of the dynamical lottery ticket hypothesis, which states that any ordinary network can be strongly compressed such that the learning dynamics and result remain unchanged. (Ib) shows that a neural scaling law of the form Lsim d^{-α} can be boosted to an arbitrarily fast power law decay, and ultimately to exp(-α' sqrt[m]{d}).

The kernel of a binary code containing the zero word partitions the binary coordinates into blocks, two coordinates lying in the same block when every kernel word takes the same value in both, and the kernel-block rank profile is the multiset of the dimensions of its linear span punctured on those blocks. For the family H^{t_1,t_2,t_3} of Z_2Z_4Z_8-linear Hadamard codes, this invariant is known explicitly and gives a complete classification of the family. In this paper, we compute it for the Z_2Z_4-linear and Z_{2^s}-linear Hadamard families with which those codes are compared, and we prove that it is constant for every nonlinear Z_2Z_4-linear Hadamard code and for every nonlinear Z_{2^s}-linear Hadamard code bar H^{a_1,dots,a_s} with sgeq2. The second statement is obtained without any rank formula, by exhibiting coordinate permutations that preserve the code and act transitively on its kernel blocks, which makes the argument uniform in s. We also prove a descent theorem: if bar H^{a_1,dots,a_s} is nonlinear and a_1geq2, then the code punctured on one kernel block is the Z_{2^{s-1}}-linear Hadamard code bar H^{a_1,dots,a_{s-1}}. Consequently, the constant local rank equals rank(bar H^{a_1,dots,a_{s-1}}) and is at least t-κ+2, where 2^t is the length and κ the kernel dimension. These results separate every nonlinear Z_2Z_4Z_8-linear Hadamard code from every Z_4-linear, Z_2Z_4-linear and Z_{2^s}-linear Hadamard code of the same length, except for the single infinite family H^{1,1,t-4} and bar H^{2,0,t-5} with tgeq5. The members of this infinite family agree in the rank, kernel dimension and the kernel-block rank profile, but a two-block refinement separates them.

For long-range percolation on Z with translation-invariant edge kernel J, it is a classical theorem of Aizenman and Newman (1986) that the phase transition is discontinuous when J(x,y) is of order |x-y|^{-2} and that there is no phase transition at all when J(x,y)=o(|x-y|^{-2}). We prove a strengthened version of this theorem for the hierarchical lattice, where the relevant threshold is at |x-y|^{-2d} loglog |x-y| rather than |x-y|^{-2}: There is a continuous phase transition for kernels of larger order, a discontinuous phase transition for kernels of exactly this order, and no phase transition at all for kernels of smaller order. As such, |x-y|^{-2d} loglog |x-y| is essentially the only kernel that produces a discontinuous phase transition. We also prove a hierarchical analogue of the ``M^2β=1'' conjecture of Imbrie and Newman (1988), which gives an exact formula for the density of the infinite cluster at the point of discontinuous phase transition and remains open in the Euclidean setting.

Let (C, ω_{C}) be a Ricci-flat, simply connected, conical Kähler manifold. We establish a Liouville theorem for constant scalar curvature Kähler (cscK) metrics on C. The theorem asserts that any cscK metric ω satisfying the uniform bound 1{C} ω_{C} leq ωleq C ω_{C} for some Cgeq1 is equal to ω_{C} up to a holomorphic automorphism that commutes with the scaling action of the cone structure. Next, we develop a C^{0,α}-estimate for uniformly bounded Kähler metrics on a ball around the apex, using a Hölder-type seminorm inspired by Krylov. This estimate applies for small α> 0 under the assumption of uniformly bounded scalar curvature. As a corollary of this result, we show that such a Kähler metric ω is asymptotic to the Ricci-flat cone metric ω_{C}, with polynomial decay rate r^α and for sufficiently small α> 0.

It is well known that H^{m,p}(Ω) = W^{m,p}(Ω) holds for any m, n in N, p in [1, infty), and open subset Ω of R^n. Due to the essential difficulty that there exists no nontrivial translation-invariant measure in infinite dimensions, it is hard to obtain its infinite-dimensional counterparts. In this paper, using infinite-dimensional analogues of the classical techniques of truncation, boundary straightening, and partition of unity, we prove that smooth cylinder functions are dense in W^{m,p}(O). Consequently, H^{m,p}(O) = W^{m,p}(O) holds for any m in N, p in [1, infty), and open subset O of ell^2 with a suitable boundary. Moreover, in the key step of compact truncation, we also prove that the Schatten p-norm type estimates for the higher-order derivatives of the Gross convolution are sharp for p = 2.

The honeycomb and Kelvin problems ask how space can be partitioned into cells of equal volume with the least surface area. We resolve this question in four dimensions for convex bodies that tile space by translations, showing that the regular 24-cell uniquely minimizes surface area at fixed volume. We develop a framework combining an exact deficit identity, valid in every dimension, with a descent argument. As a consequence, the regular 24-cell also minimizes surface area among such tiles containing a fixed ball.

We prove a new existence theorem for proper solutions of Huisken and Ilmanen's weak inverse mean curvature flow, assuming certain non-degeneracy conditions on the isoperimetric profile. In particular, no curvature assumption is imposed in our existence theorem.

We study whether an aperiodic hierarchy can provide a structural advantage for lossless compression over periodic alternatives. We show that Fibonacci quasicrystal tilings avoid the finite-depth collapse that affects periodic hierarchies: usable n-gram lookup positions remain non-zero at every level, while periodic tilings collapse after O(log p) levels for period p. This yields an aperiodic hierarchy advantage: dictionary reuse remains available across all scales instead of vanishing beyond a finite depth. Our analysis gives four main consequences. First, the Golden Compensation property shows that the exponential decay in the number of positions is exactly balanced by the exponential growth in phrase length, so potential coverage remains scale-invariant with asymptotic value Wvarphi/5. Second, using the Sturmian complexity law p(n)=n+1, we show that Fibonacci/Sturmian hierarchies maximize codebook coverage efficiency among binary aperiodic tilings. Third, under long-range dependence, the resulting hierarchy achieves lower coding entropy than comparable periodic hierarchies. Fourth, redundancy decays super-exponentially with depth, whereas periodic systems remain locked at the depth where collapse occurs. We validate these results with Quasicryth, a lossless text compressor built on a ten-level Fibonacci hierarchy with phrase lengths {2,3,5,8,13,21,34,55,89,144}. In controlled A/B experiments with identical codebooks, the aperiodic advantage over a Period-5 baseline grows from 36{,}243 B at 3 MB to 11{,}089{,}469 B at 1 GB, explained by the activation of deeper hierarchy levels. On enwik9, Quasicryth achieves 225{,}918{,}349 B (22.59%), with 20{,}735{,}733 B saved by the Fibonacci tiling relative to no tiling.

Lossless data compression remains central to computer science, with direct impact on storage, communication bandwidth, computational cost, and energy consumption. It is also closely related to Algorithmic Information Theory, where compressibility provides an operational measure of structure and non-randomness. This paper presents the 2026 Algorithmic Information Theory Data Compression Challenge, a benchmark for evaluating general-purpose lossless compressors under realistic constraints. Submissions were encouraged to use arithmetic or range coding, limited to at most 8 GB of memory, and required to include a decompressor no larger than 1 MB. The benchmark comprised sixteen heterogeneous files, split into public training and hidden testing datasets. In total, 117 valid submitted compressors were evaluated alongside established reference compressors using compression ratio, compression and decompression time, Weissman score, and Pareto-frontier analysis. The results show that performance depends strongly on the optimization criterion: fast compressors achieved the best speed-oriented scores, whereas modelling-intensive compressors produced smaller outputs at higher computational cost. A Normalized Compression Distance analysis further revealed clusters of related submissions and distinguished incremental variants from more independent implementations. Selected submissions were described for their methodological novelty or competitive performance and further tested on four large external datasets, where several achieved competitive or superior results relative to established compressors. Overall, the challenge confirms the importance of probabilistic modelling, hidden testing, and external datasets for assessing compression performance and generalization. Benchmark resources, leaderboard data, binaries, and selected source code are publicly available at https://aitdcc.github.io.

We study the two-dimensional periodic resistive compressible magnetohydrodynamic equations with constant shear viscosity and magnetic diffusivity, bulk viscosity ρ^β, and pressure ρ^γ. For every β>1 and γ>1, we construct global finite-energy renormalized weak solutions from strictly positive bounded W^{1,q} densities, q>2, and arbitrary H^1 velocity and magnetic data. The construction combines approximation-independent finite density moments, generalized effective-flux identities, and a joint logarithmic renormalization of the density and its entropy defect. An augmented relative energy containing the β-power density potential yields weak--strong uniqueness against positive H^2 strong solutions for the same full pressure range. We also establish global strong existence for a large-background family when γ>max{5-2β,β+1}. Writing m=intρ_0, the density perturbation is bounded in the scaled norm m^{(γ-3)/2}|ρ_0-m|_{H^2}, while the fixed H^2 velocity and magnetic norms need not be small. For sufficiently large m, frequency-dependent acoustic dissipation, staged vorticity weights, and a separate initial-layer flux estimate close a simultaneous bootstrap. The admissible pressure exponents have infimum 7/3, with the endpoint excluded. For β>2 the relevant damping rate is the minimum of the viscous and pressure-relaxation rates; keeping both rates extends the argument without an upper restriction on β. The strong theorem has no upper restriction on γ under this scaled density hypothesis; for fixed additive density perturbations it applies in the stated lower range up to γ=3.

A lattice quantizer approximates an arbitrary real-valued source vector with a vector taken from a specific discrete lattice. The quantization error is the difference between the source vector and the lattice vector. In a classic 1996 paper, Zamir and Feder show that the globally optimal lattice quantizer (which minimizes the mean square error) has white quantization error: for a uniformly distributed source, the covariance of the error is the identity matrix, multiplied by a positive real factor. We generalize the theorem, showing that the same property holds (i) for any lattice whose mean square error cannot be decreased by a small perturbation of the generator matrix, and (ii) for an optimal product of lattices that are themselves locally optimal in the sense of (i). We derive an upper bound on the normalized second moment (NSM) of the optimal lattice in any dimension, by proving that any lower- or upper-triangular modification to the generator matrix of a product lattice reduces the NSM. Using these tools and employing the best currently known lattice quantizers to build product lattices, we construct improved lattice quantizers in dimensions 13 to 15, 17 to 23, and 25 to 48. In some dimensions, these are the first reported lattices with normalized second moments below the best known upper bound.

Optimal transport (OT) theory has been been used in machine learning to study and characterize maps that can push-forward efficiently a probability measure onto another. Recent works have drawn inspiration from Brenier's theorem, which states that when the ground cost is the squared-Euclidean distance, the ``best'' map to morph a continuous measure in P(Rd) into another must be the gradient of a convex function. To exploit that result, [Makkuva+ 2020, Korotin+2020] consider maps T=nabla f_theta, where f_theta is an input convex neural network (ICNN), as defined by Amos+2017, and fit theta with SGD using samples. Despite their mathematical elegance, fitting OT maps with ICNNs raises many challenges, due notably to the many constraints imposed on theta; the need to approximate the conjugate of f_theta; or the limitation that they only work for the squared-Euclidean cost. More generally, we question the relevance of using Brenier's result, which only applies to densities, to constrain the architecture of candidate maps fitted on samples. Motivated by these limitations, we propose a radically different approach to estimating OT maps: Given a cost c and a reference measure rho, we introduce a regularizer, the Monge gap M^c_{rho}(T) of a map T. That gap quantifies how far a map T deviates from the ideal properties we expect from a c-OT map. In practice, we drop all architecture requirements for T and simply minimize a distance (e.g., the Sinkhorn divergence) between Tsharpmu and nu, regularized by M^c_rho(T). We study M^c_{rho}, and show how our simple pipeline outperforms significantly other baselines in practice.

Why does the low dimensionality of representations, typically dapprox 1000, not prevent modern embedding-based retrieval models from scaling to billions, or even trillions, of data points? To answer this question, we study maximal-margin embeddings in the following retrieval model, classically studied in communication complexity [PS86] and more recently in embedding-based retrieval [WBNL26]. Let Ain {0,1}^{Ntimes n} be a matrix indicating whether each of N queries is relevant to each of n documents. We are interested in the largest margin m>0, denoted by m^{rd}(d, A), for which there exist unit norm embeddings of the queries and documents {U_j}_{j = 1}^N, {V_i}_{i = 1}^n with the following property. langle U_j, V_irangle ge m whenever A_{ji} = 1 and langle U_j, V_irangle le -m otherwise. A large margin is a key proxy for representation quality: it controls both robustness to perturbations and compositional generalization across queries. Our main theorem establishes that the best possible margin without a restriction on the dimension, m^{rd}(+infty, A), can be nearly achieved in dimension d = O(m^{rd}(+infty, A)^{-2}log n) which improves a theorem of [BDES02]. Together with a matching lower bound in Theorem 1.5, we conclude that when Ain {0,1}^{n{k}times n} is the matrix containing all possible k-sparse rows once, dimension d = O(klog (n/k)) is necessary and sufficient for the maximal possible margin m^{rd}(+infty, A) = Θ(k^{-1/2}) in this setting. This fully resolves the setup of [WBNL26]. We also give several constructions for large margins when d = o(klog (n/k)). Finally, we empirically test the InfoNCE and sigmoid losses for producing large margin embeddings and demonstrate a clear advantage of the sigmoid loss.

Accurate digital surface models (DSMs) are essential for many geospatial applications, including urban monitoring, environmental analyses, infrastructure management, and change detection. However, large-scale DSMs frequently contain incomplete or outdated regions due to acquisition limitations, reconstruction artifacts, or changes in the built environment. Traditional height completion approaches primarily rely on spatial interpolation or which assume spatial continuity and therefore fail when objects are missing. Recent learning-based approaches improve reconstruction quality but typically require supervised training on sensor-specific datasets, limiting their generalization across domains and sensing conditions. We propose Prior2DSM, a training-free framework for metric DSM completion that operates entirely at test time by leveraging foundation models. Unlike previous height completion approaches that require task-specific training, the proposed method combines self-supervised Vision Transformer (ViT) features from DINOv3 with monocular depth foundation models to propagate metric information from incomplete height priors through semantic feature-space correspondence. Test-time adaptation (TTA) is performed using parameter-efficient low-rank adaptation (LoRA) together with a lightweight multilayer perceptron (MLP), which predicts spatially varying scale and shift parameters to convert relative depth estimates into metric heights. Experiments demonstrate consistent improvements over interpolation based methods, prior-based rescaling height approaches, and state-of-the-art monocular depth estimation models. Prior2DSM reduces reconstruction error while preserving structural fidelity, achieving up to a 46% reduction in RMSE compared to linear fitting of MDE, and further enables DSM updating and coupled RGB-DSM generation.

Let λ_{mathbb K}(m) denote the maximal absolute projection constant among m-dimensional Banach spaces over mathbb K=mathbb R or mathbb C. Its exact value is known only in a few cases, and determining it remains a challenging problem. In this note, we investigate several structured vector configurations that naturally arise in this context. We first recall the connection between maximal projection constants and tight frames, then consider biangular tight frames, and show that their relative and quasimaximal projection constants coincide. A particularly interesting example is provided by the midpoints of the edges of a regular simplex, which yield natural lower bounds for λ_{mathbb R}(m). Numerical evidence suggests that these bounds may be sharp in dimensions 6 and 8. We also discuss weighted spherical (2,2)-designs with a small number of vectors and their connection with maximal projection constants. Examples in dimensions 4 and 5 indicate that the sign patterns of their Gram matrices may play an important role. These observations lead us to formulate several conjectures concerning maximal absolute projection constants and the vector configurations associated with them.

A new information theoretic condition is presented for reconstructing a discrete random variable X based on the knowledge of a set of discrete functions of X. The reconstruction condition is derived from Shannon's 1953 lattice theory with two entropic metrics of Shannon and Rajski. Because such a theoretical material is relatively unknown and appears quite dispersed in different references, we first provide a synthetic description (with complete proofs) of its concepts, such as total, common and complementary informations. Definitions and properties of the two entropic metrics are also fully detailed and shown compatible with the lattice structure. A new geometric interpretation of such a lattice structure is then investigated that leads to a necessary (and sometimes sufficient) condition for reconstructing the discrete random variable X given a set { X_1,ldots,X_{n} } of elements in the lattice generated by X. Finally, this condition is illustrated in five specific examples of perfect reconstruction problems: reconstruction of a symmetric random variable from the knowledge of its sign and absolute value, reconstruction of a word from a set of linear combinations, reconstruction of an integer from its prime signature (fundamental theorem of arithmetic) and from its remainders modulo a set of coprime integers (Chinese remainder theorem), and reconstruction of the sorting permutation of a list from a minimal set of pairwise comparisons.

We explicitly construct a pair of immersed tori in three dimensional Euclidean space that are related by a mean curvature preserving isometry. These Bonnet pair tori are the first examples of compact Bonnet pairs. This resolves a longstanding open problem on whether the metric and mean curvature function determine a unique smooth compact immersion. Moreover, we prove these isometric tori are real analytic. This resolves a second longstanding open problem on whether real analyticity of the metric already determines a unique compact immersion. Our construction uses the relationship between Bonnet pairs and isothermic surfaces. The Bonnet pair tori arise as conformal transformations of an isothermic torus with one family of planar curvature lines. We classify such isothermic tori in our companion paper (arXiv:2312.14956). The above approach stems from computational investigations of a 5x7 quad decomposition of a torus using a discrete differential geometric analog of isothermic surfaces and Bonnet pairs.

Commuting families of contractions or contractive C_{0}-semigroups on Hilbert spaces often fail to admit power dilations resp, simultaneous unitary dilations which are themselves commutative (see [45, 13, 15]). In the non-commutative setting, Sz.-Nagy [60] and Bożejko [5] provided means to dilate arbitrary families of contractions. The present work extends these discrete-time results to families {T_{i}}_{i in I} of contractive C_{0}-semigroups. We refer to these dilations as continuous-time free unitary dilations and present three distinct approaches to obtain them: 1) An explicit derivation applicable to semigroups that arise as interpolations; 2) A full proof with an explicit construction, via the theory of co-generators à la Słociński [54, 55]; and 3) A second full proof based on the abstract structure of semigroups, which admits a natural reformulation to semigroups defined over topological free products of R_{geq 0} and leads to various residuality results. In 2) a IInd free dilation theorem for topologised index sets is developed via a reformulation of the Trotter--Kato theorem for co-generators. As an application of this we demonstrate how evolution families can be reduced to continuously monitored processes subject to temporal change, à la the quantum Zeno effect [22, 23, 24, 30, 37].

Model order reduction has been extensively studied over the last two decades. Projection-based methods such as the Proper Orthogonal Decomposition and the Reduced Basis Method enjoy the important advantages of Galerkin methods in the derivation of the reduced problem, but are limited to linear data compression for which the reduced solution is sought as a linear combination of spatial modes. Nonlinear data compression must be used when the solution manifold is not embedded in a low-dimensional subspace. Early methods involve piecewise linear data compression, by constructing a dictionary of reduced-order models tailored to a partition of the solution manifold. In this work, we introduce the concept of optimal partition of the solution manifold in terms of normalized Kolmogorov widths, and prove that the optimal partitions can be found by means of a representative-based clustering algorithm using the sine dissimilarity measure on the solution manifold.

We rigorously state the connection between the EHZ-capacity of convex Lagrangian products Ktimes TsubsetR^ntimesR^n and the minimal length of closed (K,T)-Minkowski billiard trajectories. This connection was made explicit for the first time by Artstein-Avidan and Ostrover under the assumption of smoothness and strict convexity of both K and T. We prove this connection in its full generality, i.e., without requiring any conditions on the convex bodies K and T. This prepares the computation of the EHZ-capacity of convex Lagrangian products of two convex polytopes by using discrete computational methods.

Compression is fundamental to intelligence. A model that can represent its training data as a short code has discovered regularities that enable generalization. Large neural networks may learn functions far simpler than their parameter counts suggest, but it is challenging to construct codes that realize this simplicity. Parameter-based methods such as quantization produce code lengths that scale with model size, insensitive to how much information the parameters store. Prequential coding bypasses this issue by compressing the training trajectory, but codes the exact data sequence regardless of how much the model learns, yielding large codes when the data has high entropy. We introduce requential coding, where a teacher model selects training samples drawn from the student's own distribution. The student's code records only these selections, which cost bits only where teacher and student disagree. The resulting code length is independent of parameter count and data entropy, and often orders of magnitude shorter than the prequential counterpart, with an advantage that grows with scale. This compression sheds light on phenomena inaccessible to prior compressors. Holding loss fixed, larger models and ensembles compress to much smaller sizes despite more parameters. Plugged into a PAC-Bayes bound, the requential code yields state-of-the-art generalization guarantees for billion-parameter LLMs, outperforming bounds built on aggressive post-training quantization even granted zero error. The bound tightens with scale in the compute-optimal regime, as models become increasingly compressible relative to dataset size. The same code predicts that models gradually overfit when trained for multiple epochs. It also isolates the learnable information in a dataset from its unpredictable, random content, revealing that lower-entropy text holds far more learnable structure than higher-entropy image data.

Quantizing large language models has become a standard way to reduce their memory and computational costs. Typically, existing methods focus on breaking down the problem into individual layer-wise sub-problems, and minimizing per-layer error, measured via various metrics. Yet, this approach currently lacks theoretical justification and the metrics employed may be sub-optimal. In this paper, we present a "linearity theorem" establishing a direct relationship between the layer-wise ell_2 reconstruction error and the model perplexity increase due to quantization. This insight enables two novel applications: (1) a simple data-free LLM quantization method using Hadamard rotations and MSE-optimal grids, dubbed HIGGS, which outperforms all prior data-free approaches such as the extremely popular NF4 quantized format, and (2) an optimal solution to the problem of finding non-uniform per-layer quantization levels which match a given compression constraint in the medium-bitwidth regime, obtained by reduction to dynamic programming. On the practical side, we demonstrate improved accuracy-compression trade-offs on Llama-3.1 and 3.2-family models, as well as on Qwen-family models. Further, we show that our method can be efficiently supported in terms of GPU kernels at various batch sizes, advancing both data-free and non-uniform quantization for LLMs.

Fix a strip width \(ρ>0\), a Diophantine frequency \(α\), an integer \(N\ge2\), and \(λ>1\). For the family \[ V_{\varepsilon,δ}(x) =2λ\cos(2πN x)+\varepsilonδ(x), \] we prove that there is an open prevalent set \(\mathcal P_{\rm strat}\) of real-analytic strip-\(ρ\) perturbations such that every \(δ\in\mathcal P_{\rm strat}\) admits \(\varepsilon_0(δ)>0\) for which \[ gathered Leb\left\{ E\inΣ(V_{\varepsilon,δ},α): ω(α,E;V_{\varepsilon,δ})=s \right\}>0, 0<\varepsilon<\varepsilon_0(δ), \qquad s=1,\ldots,N. gathered \] The proof relates acceleration to the total multiplicity of real zeros of analytic overlap coefficients for first-return cocycles. Under quantitative control of the return products, Poisson integration and Kac normalization identify acceleration with half this multiplicity. A positive analytic dual state gives the finite-volume spectral gaps and Schur curvature needed at the upper spectral edge. Schur variations separate the perturbed maxima, and parameter exclusion preserves \(s\) pairs of simple zeros on a positive-measure spectral set.

Data used for analytics and machine learning often take the form of tables with categorical entries. We introduce a family of lossless compression algorithms for such data that proceed in four steps: (i) Estimate latent variables associated to rows and columns; (ii) Partition the table in blocks according to the row/column latents; (iii) Apply a sequential (e.g. Lempel-Ziv) coder to each of the blocks; (iv) Append a compressed encoding of the latents. We evaluate it on several benchmark datasets, and study optimal compression in a probabilistic model for that tabular data, whereby latent values are independent and table entries are conditionally independent given the latent values. We prove that the model has a well defined entropy rate and satisfies an asymptotic equipartition property. We also prove that classical compression schemes such as Lempel-Ziv and finite-state encoders do not achieve this rate. On the other hand, the latent estimation strategy outlined above achieves the optimal rate.

The relationship between balance laws and the Principle of Virtual Work as well as the structure of contact interactions in continua remain foundational issues in Mechanics. In this work, we revisit these issues within the distributional framework emphasized by Paul Germain. We show that while the Principle of Virtual Work implies balance of forces and moments for nth-gradient continua, balance laws alone do not suffice to characterize equilibrium for n geq 2. We then reexamine Noll's classical theorem asserting that surface contact forces depend solely on the unit normal to the surface and identify the precise role of his additional assumptions, namely the absence of edge and wedge contact interactions and the boundedness of the surface contact density on the space of oriented surfaces. We demonstrate that these hypotheses fail for general higher-gradient continua. Consequently, the presence of curvature-dependent surface contact forces in such materials does not conflict with Noll's theorem and refutes his claim of their nonexistence.

We study a paradigm of coding for compression of the natural numbers via the zeta distribution and develop a statistical-mechanical interpretation, both in terms of Hagedorn systems and a Bose gas with energy levels given by logarithms of prime numbers. We also propose a simple coding scheme for the zeta distribution that nearly achieves the ideal code length. For block coding of vectors of natural numbers, we derive the micro-canonical entropy function and demonstrate its asymptotic linearity implying that its behavior is analogous to that of a Hagedorn system. We also derive the large deviations rate function, and provide a formula for the best coding parameter in the large deviations sense. We show that due the Hagedorn-type phase transition there is only partial equivalence of ensembles, due to the degeneration of the domain of the partition function.

Soft theorems describe the behavior of scattering amplitudes when one or several external particles are taken to be energetically soft. In tree-level gravity there are universal soft theorems for the three leading orders in the soft expansion, and they can be shown to be equivalent to Ward identities of asymptotic symmetries. While the leading and subleading symmetries are understood as supertranslations and superrotations respectively, the precise symmetry interpretation of the sub-subleading soft theorem is still a matter of investigation. The form of the sub-subleading soft graviton theorem was elucidated by Cachazo and Strominger using a BCFW expansion of graviton amplitudes. In this work we show that consistency with results based on asymptotic charges requires a careful treatment of collinear singularities in the amplitude, giving rise to collinear corrections to the usual Cachazo-Strominger soft theorem.

Many real-world machine learning problems involve inherently hierarchical data, yet traditional approaches rely on Euclidean metrics that fail to capture the discrete, branching nature of hierarchical relationships. We present a theoretical foundation for machine learning in p-adic metric spaces, which naturally respect hierarchical structure. Our main result proves that an n-dimensional plane minimizing the p-adic sum of distances to points in a dataset must pass through at least n + 1 of those points -- a striking contrast to Euclidean regression that highlights how p-adic metrics better align with the discrete nature of hierarchical data. As a corollary, a polynomial of degree n constructed to minimise the p-adic sum of residuals will pass through at least n + 1 points. As a further corollary, a polynomial of degree n approximating a higher degree polynomial at a finite number of points will yield a difference polynomial that has distinct rational roots. We demonstrate the practical significance of this result through two applications in natural language processing: analyzing hierarchical taxonomies and modeling grammatical morphology. These results suggest that p-adic metrics may be fundamental to properly handling hierarchical data structures in machine learning. In hierarchical data, interpolation between points often makes less sense than selecting actual observed points as representatives.

Deep neural networks have delivered remarkable performance and have been widely used in various visual tasks. However, their huge size causes significant inconvenience for transmission and storage. Many previous studies have explored model size compression. However, these studies often approach various lossy and lossless compression methods in isolation, leading to challenges in achieving high compression ratios efficiently. This work proposes a post-training model size compression method that combines lossy and lossless compression in a unified way. We first propose a unified parametric weight transformation, which ensures different lossy compression methods can be performed jointly in a post-training manner. Then, a dedicated differentiable counter is introduced to guide the optimization of lossy compression to arrive at a more suitable point for later lossless compression. Additionally, our method can easily control a desired global compression ratio and allocate adaptive ratios for different layers. Finally, our method can achieve a stable 10times compression ratio without sacrificing accuracy and a 20times compression ratio with minor accuracy loss in a short time. Our code is available at https://github.com/ModelTC/L2_Compression .

Motivated by applications in Diophantine approximation, we develop a new fibering mechanism for non-degenerate manifolds in R^n, showing that they admit local decompositions into families of non-degenerate curves with uniform quantitative non-degeneracy properties. The manifolds are required to be C^N with the essentially optimal bound Ngeq maxlbrace n,l+1rbrace, where the manifold is l non-degenerate. Furthermore, we show that this uniform quantitative local control can be globalised into a qualitative statement, so that almost every point of a non-degenerate manifold lies on a non-degenerate curve contained in the manifold. Our approach is perturbative and shows that the fibering structure arising in the analytic setting is stable under sufficiently small C^N perturbations. The method provides a framework for extending fibering based results in Diophantine approximation from curves to manifolds. Such applications are considered in the paper.

Idempotence is the stability of image codec to re-compression. At the first glance, it is unrelated to perceptual image compression. However, we find that theoretically: 1) Conditional generative model-based perceptual codec satisfies idempotence; 2) Unconditional generative model with idempotence constraint is equivalent to conditional generative codec. Based on this newfound equivalence, we propose a new paradigm of perceptual image codec by inverting unconditional generative model with idempotence constraints. Our codec is theoretically equivalent to conditional generative codec, and it does not require training new models. Instead, it only requires a pre-trained mean-square-error codec and unconditional generative model. Empirically, we show that our proposed approach outperforms state-of-the-art methods such as HiFiC and ILLM, in terms of Fr\'echet Inception Distance (FID). The source code is provided in https://github.com/tongdaxu/Idempotence-and-Perceptual-Image-Compression.

We study a variant of the equidistribution of mass conjecture on the sphere posed by Böcherer, Sarnak, and Schulze-Pillot: quantum unique ergodicity in shrinking sets. Conditionally on the generalized Lindelöf hypothesis, we show that quantum unique ergodicity holds on every shrinking spherical cap whose radius is considerably larger than the Planck scale, and that it holds on almost every shrinking spherical cap whose radius is larger than the Planck scale. Additionally, conditionally on GLH, we provide explicit upper bounds for the 1-Wasserstein distance and the spherical cap discrepancy between the involved measures.

We introduce the Subspace Power Method (SPM) for calculating the CP decomposition of low-rank real symmetric tensors. This algorithm calculates one new CP component at a time, alternating between applying the shifted symmetric higher-order power method (SS-HOPM) to a certain modified tensor, constructed from a matrix flattening of the original tensor; and using appropriate deflation steps. We obtain rigorous guarantees for SPM regarding convergence and global optima for input tensors of dimension d and order m of CP rank up to O(d^{lfloor m/2rfloor}), via results in classical algebraic geometry and optimization theory. As a by-product of our analysis we prove that SS-HOPM converges unconditionally, settling a conjecture in [Kolda, T.G., Mayo, J.R.: Shifted power method for computing tensor eigenpairs. SIAM Journal on Matrix Analysis and Applications 32(4), 1095-1124 (2011)]. We present numerical experiments which demonstrate that SPM is efficient and robust to noise, being up to one order of magnitude faster than state-of-the-art CP decomposition algorithms in certain experiments. Furthermore, prior knowledge of the CP rank is not required by SPM.

We provide a rigorous derivation of Boltzmann's kinetic equation from the hard sphere system for rarefied gas, which is valid for arbitrarily long times, as long as the solution to the Boltzmann equation exists. This extends Lanford's landmark theorem (1975), which justifies this derivation for a sufficiently short time. In a companion paper (arXiv:2503.01800), we connect this derivation to existing literature on hydrodynamic limits. This completes the resolution of Hilbert's Sixth Problem pertaining to the derivation of fluid equations from Newton's laws, in the case of a rarefied, hard sphere gas. The general strategy follows the paradigm introduced by the first two authors for the long-time derivation of the wave kinetic equation in wave turbulence theory. This is based on propagating a long-time cumulant ansatz, which keeps memory of the full collision history of the relevant particles, by an important partial time expansion. The heart of the matter is proving the smallness of these cumulants in L^1, which can be reduced to combinatorial properties for the associated diagrams which we call molecules. These properties are then proved by devising an elaborate cutting algorithm, which is a major novelty of this work.

In this paper, it is shown that if a sequence of resistance metric spaces equipped with measures converges with respect to the local Gromov-Hausdorff-vague topology, and certain non-explosion and metric-entropy conditions are satisfied, then the associated stochastic processes and their local times also converge. The metric-entropy condition can be checked by applying volume estimates of balls. Whilst similar results have been proved previously, the approach of this article is more widely applicable. Indeed, we recover various known conclusions for scaling limits of some deterministic self-similar fractal graphs, critical Galton-Watson trees, the critical Erdos-R\'enyi random graph and the configuration model (in the latter two cases, we prove for the first time the convergence of the models with respect to the resistance metric and also, for the configuration model, we overcome an error in the existing proof of local time convergence). Moreover, we derive new ones for scaling limits of uniform spanning trees and random recursive fractals. The metric-entropy condition also implies convergence of associated Gaussian processes.

We study singularities of height functions on singular surfaces in R^3 parameterized by smooth map-germs A-equivalent to H_k in Mond's classification, and the versality of the family of the height functions. We also study relations of the singularities of the height functions with the parabolic locus of these singular surfaces.

In this paper, we find a sample complexity bound for learning a simplex from noisy samples. Assume a dataset of size n is given which includes i.i.d. samples drawn from a uniform distribution over an unknown simplex in R^K, where samples are assumed to be corrupted by a multi-variate additive Gaussian noise of an arbitrary magnitude. We prove the existence of an algorithm that with high probability outputs a simplex having a ell_2 distance of at most varepsilon from the true simplex (for any varepsilon>0). Also, we theoretically show that in order to achieve this bound, it is sufficient to have ngeleft(K^2/varepsilon^2right)e^{Omegaleft(K/SNR^2right)} samples, where SNR stands for the signal-to-noise ratio. This result solves an important open problem and shows as long as SNRgeOmegaleft(K^{1/2}right), the sample complexity of the noisy regime has the same order to that of the noiseless case. Our proofs are a combination of the so-called sample compression technique in ashtiani2018nearly, mathematical tools from high-dimensional geometry, and Fourier analysis. In particular, we have proposed a general Fourier-based technique for recovery of a more general class of distribution families from additive Gaussian noise, which can be further used in a variety of other related problems.

We present an algorithm for efficiently exploring inequivalent Calabi-Yau threefold hypersurfaces in toric varieties. A direct enumeration of fine, regular, star triangulations (FRSTs) of polytopes in the Kreuzer-Skarke database is foreseeably impossible due to the large count of distinct FRSTs. Moreover, such an enumeration is needlessly redundant because many such triangulations have the same restrictions to 2-faces and hence, by Wall's theorem, lead to equivalent Calabi-Yau threefolds. We show that this redundancy can be circumvented by finding a height vector in the strict interior of the intersection of the secondary cones associated with each 2-face triangulation. We demonstrate that such triangulations are generated with orders of magnitude fewer operations than the naive approach of generating all FRSTs and selecting only those differing on 2-faces. Similar methods are also presented to directly generate (the support of) the secondary subfan of all fine triangulations, relevant for random sampling of FRSTs.

In this paper, we introduce fundamental notions of homotopy theory, including homotopy excision and the Freudenthal suspension theorem. We then explore framed cobordism and its connection to stable homotopy groups of spheres through the Pontryagin-Thom construction. Using this framework, we compute the stable stems in dimensions 0, 1, and 2. This work is primarily expository, revisiting proofs from Pont1 with slight modifications incorporating modern notation. Furthermore, in the final section, we discuss 2-dimensional framed manifolds with Arf invariant one and examine why the result of Pont2 regarding π_2^S is incorrect.

Low Rank Decomposition of matrix - splitting a large matrix into a product of two smaller matrix offers a means for compression that reduces the parameters of a model without sparsification, and hence delivering more speedup on modern hardware. Moreover, unlike quantization, the compressed linear layers remain fully differentiable and all the parameters trainable, while being able to leverage the existing highly efficient kernels over floating point matrices. We study the potential to compress Large Language Models (LLMs) for monolingual Code generation via Low Rank Decomposition (LoRD) and observe that ranks for the linear layers in these models can be reduced by upto 39.58% with less than 1% increase in perplexity. We then use Low Rank Decomposition (LoRD) to compress StarCoder 16B to 13.2B parameter with no drop and to 12.3B with minimal drop in HumanEval Pass@1 score, in less than 10 minutes on a single A100. The compressed models speeds up inference by up to 22.35% with just a single line of change in code over huggingface's implementation with pytorch backend. Low Rank Decomposition (LoRD) models remain compatible with state of the art near-lossless quantization method such as SpQR, which allows leveraging further compression gains of quantization. Lastly, QLoRA over Low Rank Decomposition (LoRD) model further reduces memory requirements by as much as 21.2% over vanilla QLoRA while offering similar gains from parameter efficient fine tuning. Our work shows Low Rank Decomposition (LoRD) as a promising new paradigm for LLM compression.

Large language model compression has made substantial progress through pruning, quantization, and low-rank decomposition, yet a fundamental limitation persists across all existing methods: compression budgets are allocated without any representation of what individual model components functionally encode. We term this the capability-blind compression problem and argue it is a root cause of two well-documented failures -- the insensitivity of perplexity-based evaluation to reasoning capability loss, and the abrupt phase transitions in model performance recently characterized by Ma et al. (2026). We propose Capability-Guided Compression (CGC), a framework that addresses this by using Sparse Autoencoder (SAE)-derived capability density maps to allocate differential compression budgets across transformer components. Capability density is a formally defined scalar measure combining the feature breadth, activation entropy, and cross-input consistency of a component's SAE feature activation distribution. We prove theoretically that components with higher capability density exhibit lower structural redundancy and reach their individual phase transition points at lower compression ratios, providing the first pre-compression mechanism for component-level phase transition prediction. Experiments on GPT-2 Medium confirm that capability density is statistically independent of Wanda importance scores (Spearman rho = -0.054, n = 384 heads), establishing it as a genuinely novel compression signal orthogonal to all existing importance metrics. We report a negative result on PPL-based compression comparison and provide a principled diagnosis identifying GPT-2 Medium as an insufficient test bed for the full CGC hypothesis. The theoretical framework, density formalism, and orthogonality finding constitute a foundation for capability-aware compression research.

Adapting standard methods from geometric measure theory, we provide an example of a polynomial-valued measure mu on tame sets in R^d which satisfies many desirable properties. Among these is strict monotonicity: the measure of a proper subset is strictly less than the measure of the whole set. Using techniques from non-standard analysis, we display that the domain of mu can be extended to all subsets of R^d (up to equivalence modulo infinitesimals). The resulting extension is a numerosity function that encodes the i-dimensional Hausdorff measure for all iin N, as well as the i-th intrinsic volume functions.

A selector that allocates work across opaque components must evaluate them without being told how they performed. We study the least communication that suffices. Each parent maintains an allocation vector over its children and updates it from outcomes by proportional redistribution. Each child reads the sign of its own allocation change, one bit per round it is selected, so no evaluation message crosses the parent-child boundary. (1) The update preserves the simplex with strict positivity. (2) Conditional on being selected, a child's sign is exactly the root outcome at every depth, a sufficient statistic. The conditioning is necessary. (3) Under interiority a single selector has a unique interior equilibrium, exact and closed-form for N=2, with almost sure convergence under decreasing steps. For general N an equi-ratio condition gives an explicit affine equilibrium. (4) The linearised mean flow has real, strictly negative tangent eigenvalues with explicit bounds for all N >= 2. (5) The mean flow converges globally. An explicit potential, up to normalisation the KL divergence between the failure and slack distributions, decreases at a rate equal to the variance of the equi-ratio statistic. Without interiority the rule drops exactly the children below the surviving support's threshold. (6) The update law composes along a single active path, since ancestors select a node's active rounds but do not alter a realised signal. No joint convergence theorem for a hierarchy whose levels adapt simultaneously is claimed. (7) The one-bit interface admits exactly four deterministic memoryless distortions, and at a single selector passing the bit on unchanged delivers strictly higher mean quality upward than negation or constant transmission. Illustrations on synthetic hierarchies up to 16,384 leaves and three real datasets (3.4 billion comparisons, no mismatches) are consistent with the theory.

We study the Yamabe--Escobar equation prescribing positive constant scalar curvature and minimal boundary. The compactness conjecture asks whether being conformally equivalent to the round hemisphere is the only obstruction for its set of positive solutions to be compact. In this paper we show that this is true for dimensions 3leq nleq 14. We show that this dimensional threshold n_*=14 is sharp by constructing in ngeq 15 nonumbilic and non-locally-conformally-flat examples of metrics in the hemisphere for which a noncompact sequence of solutions exists. We do the same for the special classes of locally conformally flat metrics, n_*^{LCF}=18, and the metrics with umbilic boundary, n_*^{umb} =20. In all three settings we also prove that the Weyl--umbilicity vanishing conjecture holds for 4leq nleq n_* and construct counterexamples in n>n_* by perturbing the noncompactness examples. To the best of the author's knowledge, these are the first locally conformally flat noncompactness constructions for the positive-scalar-curvature, minimal-boundary case of the Yamabe--Escobar equation, outside the conformal class of the round hemisphere.

Large collections of matrices arise throughout modern machine learning, signal processing, and scientific computing, where they are commonly compressed by concatenation followed by truncated singular value decomposition (SVD). This strategy enables parameter sharing and efficient reconstruction and has been widely adopted across domains ranging from multi-view learning and signal processing to neural network compression. However, it leaves a fundamental question unanswered: which matrices can be safely concatenated and compressed together under explicit reconstruction error constraints? Existing approaches rely on heuristic or architecture-specific grouping and provide no principled guarantees on the resulting SVD approximation error. In the present work, we introduce a theory-driven framework for compression-aware clustering of matrices under SVD compression constraints. Our analysis establishes new spectral bounds for horizontally concatenated matrices, deriving global upper bounds on the optimal rank-r SVD reconstruction error from lower bounds on singular value growth. The first bound follows from Weyl-type monotonicity under blockwise extensions, while the second leverages singular values of incremental residuals to yield tighter, per-block guarantees. We further develop an efficient approximate estimator based on incremental truncated SVD that tracks dominant singular values without forming the full concatenated matrix. Therefore, we propose three clustering algorithms that merge matrices only when their predicted joint SVD compression error remains below a user-specified threshold. The algorithms span a trade-off between speed, provable accuracy, and scalability, enabling compression-aware clustering with explicit error control. Code is available online.

Recent works on compression of large language models (LLM) using quantization considered reparameterizing the architecture such that weights are distributed on the sphere. This demonstratively improves the ability to quantize by increasing the mathematical notion of coherence, resulting in fewer weight outliers without affecting the network output. In this work, we aim to further exploit this spherical geometry of the weights when performing quantization by considering Pyramid Vector Quantization (PVQ) for large language models. Arranging points evenly on the sphere is notoriously difficult, especially in high dimensions, and in case approximate solutions exists, representing points explicitly in a codebook is typically not feasible due to its additional memory cost. Instead, PVQ uses a fixed integer lattice on the sphere by projecting points onto the 1-sphere, which allows for efficient encoding and decoding without requiring an explicit codebook in memory. To obtain a practical algorithm, we propose to combine PVQ with scale quantization for which we derive theoretically optimal quantizations, under empirically verified assumptions. Further, we extend pyramid vector quantization to use Hessian information to minimize quantization error under expected feature activations, instead of only relying on weight magnitudes. Experimentally, we achieves state-of-the-art quantization performance with pareto-optimal trade-off between performance and bits per weight and bits per activation, compared to compared methods. On weight-only, we find that we can quantize a Llama-3 70B model to 3.25 bits per weight and retain 98\% accuracy on downstream tasks.

We introduce a classification conjecture for kappa-solutions in 4d Ricci flow. Our conjectured list includes known examples from the literature, but also a new 1-parameter family of Z_2^2times O_3-symmetric bubble-sheet ovals that we construct. We observe that some special cases of the conjecture follow from recent results in the literature. We also introduce a stronger variant of the classification conjecture for ancient asymptotically cylindrical 4d Ricci flows, which does not assume smoothness and nonnegative curvature operator a priori. Assuming this stronger variant holds true, we establish a canonical neighborhood theorem for 4d Ricci flow through cylindrical singularities, which shares some elements in common with Perelman's canonical neighborhood theorem for 3d Ricci flow as well as the mean-convex neighborhood theorem for mean curvature flow through neck-singularities. Finally, we argue that quotient-necks lead to new phenomena, and sketch an example of non-uniqueness for 4d Ricci flow through singularities.

LiDAR point clouds are fundamental to various applications, yet the extreme sparsity of high-precision geometric details hinders efficient context modeling, thereby limiting the compression speed and performance of existing methods. To address this challenge, we propose a compact representation for efficient predictive lossless coding. Our framework comprises two lightweight modules. First, the Geometry Re-Densification Module iteratively densifies encoded sparse geometry, extracts features at a dense scale, and then sparsifies the features for predictive coding. This module avoids costly computation on highly sparse details while maintaining a lightweight prediction head. Second, the Cross-scale Feature Propagation Module leverages occupancy cues from multiple resolution levels to guide hierarchical feature propagation, enabling information sharing across scales and reducing redundant feature extraction. Additionally, we introduce an integer-only inference pipeline to enable bit-exact cross-platform consistency, which avoids the entropy-coding collapse observed in existing neural compression methods and further accelerates coding. Experiments demonstrate competitive compression performance at real-time speed. Code will be released upon acceptance. Code is available at https://github.com/pengpeng-yu/FastPCC.

Scientific datasets present unique challenges for machine learning-driven compression methods, including more stringent requirements on accuracy and mitigation of potential invalidating artifacts. Drawing on results from compressed sensing and rate-distortion theory, we introduce effective data compression methods by developing autoencoders using high dimensional latent spaces that are L^1-regularized to obtain sparse low dimensional representations. We show how these information-rich latent spaces can be used to mitigate blurring and other artifacts to obtain highly effective data compression methods for scientific data. We demonstrate our methods for short angle scattering (SAS) datasets showing they can achieve compression ratios around two orders of magnitude and in some cases better. Our compression methods show promise for use in addressing current bottlenecks in transmission, storage, and analysis in high-performance distributed computing environments. This is central to processing the large volume of SAS data being generated at shared experimental facilities around the world to support scientific investigations. Our approaches provide general ways for obtaining specialized compression methods for targeted scientific datasets.

Let T be a rooted and weighted tree, where the weight of any node is equal to the sum of the weights of its children. The popular Treemap algorithm visualizes such a tree as a hierarchical partition of a square into rectangles, where the area of the rectangle corresponding to any node in T is equal to the weight of that node. The aspect ratio of the rectangles in such a rectangular partition necessarily depends on the weights and can become arbitrarily high. We introduce a new hierarchical partition scheme, called a polygonal partition, which uses convex polygons rather than just rectangles. We present two methods for constructing polygonal partitions, both having guarantees on the worst-case aspect ratio of the constructed polygons; in particular, both methods guarantee a bound on the aspect ratio that is independent of the weights of the nodes. We also consider rectangular partitions with slack, where the areas of the rectangles may differ slightly from the weights of the corresponding nodes. We show that this makes it possible to obtain partitions with constant aspect ratio. This result generalizes to hyper-rectangular partitions in R^d. We use these partitions with slack for embedding ultrametrics into d-dimensional Euclidean space: we give a rm polylog(Delta)-approximation algorithm for embedding n-point ultrametrics into R^d with minimum distortion, where Delta denotes the spread of the metric, i.e., the ratio between the largest and the smallest distance between two points. The previously best-known approximation ratio for this problem was polynomial in n. This is the first algorithm for embedding a non-trivial family of weighted-graph metrics into a space of constant dimension that achieves polylogarithmic approximation ratio.