arXiv subjects

Zhipeng Lu

Publications and source records attributed to Zhipeng Lu.

At least 19 recordsLinked to original sources

arXiv2026

Integral saturation of adjacency-degree lattices

Let $A$ be the adjacency matrix of $G\sim G(n,1/2)$, let $D=\operatorname{diag}(A\mathbf{1})$, and let $Λ_G=\mathbb{Z}\langle A,D\rangle\mathbf{1}$. We prove $\Pr[Λ_G\ne\mathbb{Z}^n]\le n^{-3/20+o(1)}$. Thus two dependent graph operators generate the entire vertex lattice from the constant vector, simultaneously over every prime field. Adjacency alone has a modulo-two rank obstruction for every simple graph of order greater than one, even though its rational word span is typically full. The proof first constructs words containing at most two adjacency operations whose integer span contains $Δ_G^3\mathbb{Z}^n$, where $Δ_G$ is the product of the pairwise differences between distinct degree values. This confines all possible arithmetic obstructions to a finite prime range on a single event. We then remove the remaining obstructions using conditional alternating forms, symmetric random-matrix code estimates, and uniform counts of degree-graded invariant subspaces. We connect the integer lattice to graph reconstruction through pointed moments. Saturation forces every pointed orthogonal AD intertwiner to be a vertex permutation. The short-word construction also shows that a fixed polynomial-size family of ordinary caterpillars, of diameter at most seven and order $O(\sqrt{n\log n})$, identifies almost every graph among all finite simple graphs. For nonsaturated full-rank lattices, we give an exact description of the moment fibre by admissible unimodular overlattices; at index two, the only possible ambiguity is a specified four-vertex Godsil--McKay switch.

math.CO↗

arXiv2026

Uniform adjacency-degree probes from differential rigidity

We construct a fixed triangular family of noncommutative polynomials that spans the joint cyclic module of a real symmetric pair from any prescribed vector. In dimension at most $N$, when the second matrix has spectral diameter at most $N-1$, the family has $N(N+1)/2$ members, degree at most $N-1$, and a shared arithmetic circuit of size $O(N^2)$. No rank-dependent choice of coefficients is made. The proof uses a rational differential connection with a small self-adjoint residue: a primitive polynomial exterior vector for an invariant subspace must have an integer degree smaller than one. For connected trees, the probes and the pointed adjacency--degree reconstruction theorem give one explicit self-adjoint polynomial $p_N\in\mathbb{Z}\langle a,d\rangle$ whose trace separates all trees of order at most $N$. Its circuit has $O(N^2)$ gates, formal degree at most $3N-2$, and polynomial bit complexity. Only the original adjacency and degree matrices, the identity, and ordinary matrix arithmetic are used. A separate integer sum-of-squares construction gives a fixed positive semidefinite representative on which the all-one vector is cyclic; its degree is $O(N^2)$. A linear trace-degree lower bound shows that the first construction has the correct asymptotic degree scale.

math.CO↗

arXiv2026

Explicit matrix hitting points for noncommutative algebraic branching programs

We give a deterministic polynomial-resource black-box identity test for noncommutative algebraic branching programs (ABPs), using ordinary matrix evaluations of the output polynomial. Given an alphabet of size $m\ge1$ and an ABP vertex bound $S\ge1$, we construct one fixed tuple of integer matrices of order $R=(m+2)d(d+1)/2$, where $d=(m+2)S$, with entry bit length $O(d\log(d+1))$. Every nonzero polynomial computed by an ABP with at most $S$ vertices has a nonzero evaluation at this tuple. The construction accesses neither the hidden ABP nor its intermediate gates. A rectangular variant improves the order to $O(m^{3/2}S^2)$. Matrix order, description length, and all returned entries are counted; the total resources are polynomial. The result holds in characteristic zero and in characteristic $p>\binom d2$. The main ingredient is a fixed polynomial frame spanning the joint cyclic module of every matrix tuple from every prescribed vector. A compensating block makes a tagged encoding nilpotent of index at most three, while preserving the original cyclic module. A differential exterior-vector argument then gives a single triangular recurrence with polynomially many outputs. Its shared left-linear circuit is converted into actual integer query matrices by coefficient duality. Two-sided pairing further gives deterministic nonadaptive reconstruction of an equivalent ABP from polynomially many output-only matrix queries. The same frame gives polynomial-cardinality explicit trace separators for simultaneous-conjugation orbit closures, and a fixed nilpotent substitution for bounded-dimensional recognizable series. The latter also tests rational formulas whose given syntax is defined at the scalar origin. We provide complete proofs, construction costs, and exact implementation checks.

math.RA↗

arXiv2026

On Completion Times under Memoryless Catastrophe

We study the completion time of a task subject to independent reset (catastrophe) at each step. The completion-time PGF depends on the base-process PGF through an affine relation, and we exploit this structure systematically. Our main result shows that, among age-based catastrophe mechanisms, geometric-tail catastrophe is exactly the class that yields uniform affine PGF structure; in continuous time, the characterization sharpens to Poisson resetting. We establish a sharp two-sided Kolmogorov bound of order $p+|α|$ for the exponential approximation $d_K(T/E[T], \mathrm{Exp}(1))$, thereby closing a logarithmic gap. Applications to the coupon collector with reset coupons reveal a discontinuous Gumbel-to-Exponential transition under resetting, while a multi-phase model exhibits a Gaussian-to-exponential transition with exponential convergence rate.

math.PR↗

arXiv2026

Retained-Set Descent for Diagonal Ramsey Numbers

We study how far a fixed Ramsey upper bound can be improved by descending through blue neighborhoods in one vertex set while keeping a second set fixed. A weighted inequality in the two set sizes determines when the descent can stop. For the source bound specified here, the infimum diagonal exponent over all finite derivations lies in $[1.305,\,1.307]$. A finite derivation gives $R(k,k)\le3.69507^k$ for all sufficiently large $k$; a concave polygon proves the lower bound for every finite depth. We also characterize the infimum as a greatest fixed point and show that every larger exponent has a finite derivation valid uniformly for nearby clique-size ratios.

math.CO↗

arXiv2026

Spectral extremes under exact cycle conditioning

Let $P_n$ be the matrix of a random permutation of $n$ symbols and let $M_n=\log\max_{|z|=1}|\det(I-zP_n)|$. Cook and Zeitouni proved that $M_n/\log n$ converges in probability to a constant $x_0$ for a uniform permutation. We show that the $\sqrt{\log n}$ fluctuations of $M_n$ are carried entirely by the number of cycles $K_n$. Write $λ(s)=\log\{Γ(1+s)/Γ(1+s/2)^2\}$, let $s_κ$ minimize $(1+κλ(s))/s$ on $(0,\infty)$, and put $v(κ)=κλ'(s_κ)$ and $a_θ=λ(s_θ)/s_θ$. Under the Ewens measure with any fixed parameter $θ>0$ we prove $M_n=v(θ)\log n+a_θ(K_n-θ\log n)+O_P(\log\log n)$, so that the standardized pair $(K_n,M_n)$ converges jointly to $(G,G)$ with $G$ standard normal: the maximum and the cycle count are asymptotically perfectly aligned. This is deduced from a statement about the exact conditional law, which does not depend on $θ$: for every compact $[κ_-,κ_+]\subset(0,\infty)$ there is a finite $C$ such that $P(|M_n-v(k/\log n)\log n|>C\log\log n \mid K_n=k)$ tends to $0$ uniformly over integers $k$ with $κ_-\log n\le k\leκ_+\log n$, that is, over exact and possibly atypical cycle counts. The proof keeps the size and the cycle count simultaneously in a two-variable coefficient extraction. Cycles longer than $n/(\log n)^4$ are reserved as an analytic factor whose coefficients are flat under every size shift produced by the shorter cycles; positivity then converts a scalar coefficient asymptotic into a relative comparison of the entire path-constrained measure, with an error that does not degrade with the number of constraints or with the rarity of the event. The constrained lower bound comes from pointwise saddle estimates for killed convolutions along a dyadic chain of endpoint boxes.

math.CO↗

arXiv2026

A dot-product bound from separate growth-minimizing bases

For every finite $P\subset\mathbb{R}^2$ we prove $|\{p\cdot q: p,q\in P\}|\gg |P|^{199/295}$, with an absolute constant and no logarithmic loss, where $199/295 = 2/3+7/885$. This improves the bound $2/3+7/1425$ of Kokkinos (arXiv:2502.12727), which in turn had improved the first superthreshold bound of Hanson, Roche-Newton, and Senger. The main new ingredient is the inequality $|F^{(2)}G^{(2)}/(F^{(2)}G)| \le |AB|(|AF|/|A|)^4(|BG|/|B|)^3$: each radial profile retains its own Petridis growth-minimizing subset, and the product of the two subsets serves as a common base for both growth operators; a prime-power construction shows this is sharp under its hypotheses. The second ingredient is a weighted-median replacement for the dyadic selection in the squeezing argument of Roche-Newton and Wong, upgrading their seven-factor expander to a logarithm-free form; an appendix gives the complete proof from the Solymosi-Zahl incidence theorem. We complement the lower bound with a third-moment structure theorem for near-extremal configurations and a construction showing that the pinned problem admits no linear lower bound: $\max_{p\in P}|p\cdot P| = O(|P|/\sqrt{\log |P|})$ is attainable. The algebraic core, the median lemma, the surface identities, and the pinned construction are formally verified in Lean 4 and included as ancillary files.

math.CO↗

arXiv2026

Punctured adjacency-degree algebras of Cartesian products

For a connected regular graph G and a vertex a, we study the algebra generated by the adjacency and degree matrices of G-a and its cyclic module P_a generated by the all-ones vector. Our main theorem determines dim P_a for Cartesian products whose factors have equitable distance partitions at the chosen roots. A normalized logarithmic derivative of the local spectral generating function partitions the factors into boundary classes. We identify the boundary-return space exactly and express dim P_a as a sum of affine ranks on additive spectral fibres. For a distance-regular factor with distinct spectrum Θ, this gives dim P_a(F^{\square m}) = |mΘ| - 1. For products of powers of two distinct complete graphs, we evaluate the fibre formula in closed form. We also determine the full punctured algebras of all Hamming graphs: equality with the compressed Terwilliger algebra holds precisely in dimensions at most four for the hypercube and at most two for larger alphabets. For distance-regular graphs, adjacency moments alone determine the intersection array, with an explicit finite reconstruction. Finally, Cartesian stabilizer formulas separate metric loss from orbit splitting; on Doob graphs their distance-graded defect recovers the number of Shrikhande factors.

math.CO↗

arXiv2026

The Price of Order in the Logarithmic Method

The logarithmic method is a classical static-to-dynamic transformation: it stores one dynamic ordered set as several immutable static components and rebuilds them by merges. The same component-and-merge discipline underlies write-optimized ordered indexes, where cheap insertions must be reconciled with exact ordered queries. In this paper, we study the insertion-only version after $n$ insertions, over abstract keys, in a strongly materialized merge-stack model with sequential component merges and one forward scan of the live components per query. We bound the product between the total amount of data written during the $n$ insertions and the worst-case amount of data read by a single query, known as the write-read product. The optimal bounds are as follows: - Membership and local certificates: $Θ(n\log^2 n)$. - Order and range queries with named keys or endpoints: $Θ(n\log^3 n)$. - Select: $Θ(n^2)$. Thus, the logarithmic method does not impose a universal dynamic overhead: under materialized one-way access, the optimum depends on what information the query reveals before the scan starts. This pinpoints the access-model obstruction behind the extra logarithm for exact order and range queries, and the quadratic barrier for select.

cs.DS↗

arXiv2026

Adjacency-degree algebras and spectral determination of graphs

McKay proved that the spectra of all polynomial functions of the adjacency matrix $A$ and the diagonal degree matrix $D$ determine a tree. We prove a principal version of this theorem. Let $\mathcal A(G)=\langle I,A_G,D_G\rangle$ and let $M_G=\mathcal A(G)\mathbf1$ be the cyclic module generated by the all-ones vector. For connected graphs the ideal $\mathcal A(G)J\mathcal A(G)$, where $J=\mathbf1\mathbf1^T$, acts on $M_G$ as the full endomorphism algebra. We show that every forest satisfies $M_G=U_G$, the automorphism-orbit module, and that the induced algebra on the orbit quotient of a tree is a full matrix algebra. It follows that the scalar moments $\mathbf1^Tw(A_T,D_T)\mathbf1$ determine every tree. For general graphs these moments are degree-decorated caterpillar homomorphism counts. The resulting moment-rigidity class lies inside the amenable, compact, refinable hierarchy of color refinement, and its first small-order failures are ten-vertex integral switchings invisible to $M_G$.

math.CO↗

arXiv2026

Depth over Fidelity in Fixed-Budget Noisy Evolution Strategies

Noisy evolution strategies under fixed evaluation budgets face a depth-fidelity trade-off: spending evaluations to denoise intra-generation rankings reduces the number of distribution updates the optimizer can execute. We argue for depth over fidelity and propose probabilistic elite membership (PEM), which replaces hard rank-based weights in evolution strategies with conditional expected rank weights that integrate over ranking uncertainty. PEM preserves the conditional mean update while reducing conditional update dispersion, a Rao-Blackwellization of the noisy rank-based step. We instantiate PEM via residual bootstrapping (RB-PEM) with capped per-generation overhead, complemented by an adaptive probe-and-switch mechanism for low-noise regimes. Across the COCO bbob-noisy suite and external tasks including RL policy search and hyperparameter optimization, RB-PEM achieves consistent gains in high-misranking, budget-constrained settings.

cs.NE↗

arXiv2025

A Set-to-Set Distance Measure in Hyperbolic Space

We propose a hyperbolic set-to-set distance measure for computing dissimilarity between sets in hyperbolic space. While point-to-point distances in hyperbolic space effectively capture hierarchical relationships between data points, many real-world applications require comparing sets of hyperbolic data points, where the local structure and the global structure of the sets carry crucial semantic information. The proposed the \underline{h}yperbolic \underline{s}et-\underline{to}-\underline{s}et \underline{d}istance measure (HS2SD) integrates both global and local structural information: global structure through geodesic distances between Einstein midpoints of hyperbolic sets, and local structure through topological characteristics of the two sets. To efficiently compute topological differences, we prove that using a finite Thue-Morse sequence of degree and adjacency matrices can serve as a robust approximation to capture the topological structure of a set. In this case, by considering the topological differences, HS2SD provides a more nuanced understanding of the relationships between two hyperbolic sets. Empirical evaluation on entity matching, standard image classification, and few-shot image classification demonstrates that our distance measure outperforms existing methods by effectively modeling the hierarchical and complex relationships inherent in hyperbolic sets.

cs.CV↗

arXiv2025

Geometry-aware Distance Measure for Diverse Hierarchical Structures in Hyperbolic Spaces

Learning in hyperbolic spaces has attracted increasing attention due to its superior ability to model hierarchical structures of data. Most existing hyperbolic learning methods use fixed distance measures for all data, assuming a uniform hierarchy across all data points. However, real-world hierarchical structures exhibit significant diversity, making this assumption overly restrictive. In this paper, we propose a geometry-aware distance measure in hyperbolic spaces, which dynamically adapts to varying hierarchical structures. Our approach derives the distance measure by generating tailored projections and curvatures for each pair of data points, effectively mapping them to an appropriate hyperbolic space. We introduce a revised low-rank decomposition scheme and a hard-pair mining mechanism to mitigate the computational cost of pair-wise distance computation without compromising accuracy. We present an upper bound on the low-rank approximation error using Talagrand's concentration inequality, ensuring theoretical robustness. Extensive experiments on standard image classification (MNIST, CIFAR-10 and CIFAR-100), hierarchical classification (5-level CIFAR-100), and few-shot learning tasks (mini-ImageNet, tiered-ImageNet) demonstrate the effectiveness of our method. Our approach consistently outperforms learning methods that use fixed distance measures, with notable improvements on few-shot learning tasks, where it achieves over 5\% gains on mini-ImageNet. The results reveal that adaptive distance measures better capture diverse hierarchical structures, with visualization showing clearer class boundaries and improved prototype separation in hyperbolic spaces.

cs.CV↗

arXiv2025

Convolutional optimization with convex kernel and power lift

We focus on establishing the foundational paradigm of a novel optimization theory based on convolution with convex kernels. Our goal is to devise a morally deterministic model of locating the global optima of an arbitrary function, which is distinguished from most commonly used statistical models. Limited preliminary numerical results are provided to test the efficiency of some specific algorithms derived from our paradigm, which we hope to stimulate further practical interest.

math.OC↗

arXiv2025

Reinforcement-learning-based Algorithms for Optimization Problems and Applications to Inverse Problems

We design a new iterative algorithm, called REINFORCE-OPT, for solving a general type of optimization problems. This algorithm parameterizes the solution search rule and iteratively updates the parameter using a reinforcement learning (RL) algorithm resembling REINFORCE. To gain a deeper understanding of the RL-based methods, we show that REINFORCE-OPT essentially solves a stochastic version of the given optimization problem, and that under standard assumptions, the searching rule parameter almost surely converges to a locally optimal value. Experiments show that REINFORCE-OPT outperforms other optimization methods such as gradient descent, the genetic algorithm, and particle swarm optimization, via its ability to escape from locally optimal solutions and its robustness to the choice of initial values. With rigorous derivations, we formally introduce the use of reinforcement learning to deal with inverse problems. By choosing specific probability models for the action-selection rule, we can also connect our approach to the conventional methods of Tikhonov regularization and iterative regularization. We take non-linear integral equations and parameter-identification problems in partial differential equations as examples to show how reinforcement learning can be applied in solving non-linear inverse problems. The numerical experiments highlight the strong performance of REINFORCE-OPT, as well as its ability to quantify uncertainty in error estimates and identify multiple solutions for ill-posed inverse problems that lack solution stability and uniqueness.

math.OC↗

arXiv2024

Residual Hyperbolic Graph Convolution Networks

Hyperbolic graph convolutional networks (HGCNs) have demonstrated representational capabilities of modeling hierarchical-structured graphs. However, as in general GCNs, over-smoothing may occur as the number of model layers increases, limiting the representation capabilities of most current HGCN models. In this paper, we propose residual hyperbolic graph convolutional networks (R-HGCNs) to address the over-smoothing problem. We introduce a hyperbolic residual connection function to overcome the over-smoothing problem, and also theoretically prove the effectiveness of the hyperbolic residual function. Moreover, we use product manifolds and HyperDrop to facilitate the R-HGCNs. The distinctive features of the R-HGCNs are as follows: (1) The hyperbolic residual connection preserves the initial node information in each layer and adds a hyperbolic identity mapping to prevent node features from being indistinguishable. (2) Product manifolds in R-HGCNs have been set up with different origin points in different components to facilitate the extraction of feature information from a wider range of perspectives, which enhances the representing capability of R-HGCNs. (3) HyperDrop adds multiplicative Gaussian noise into hyperbolic representations, such that perturbations can be added to alleviate the over-fitting problem without deconstructing the hyperbolic geometry. Experiment results demonstrate the effectiveness of R-HGCNs under various graph convolution layers and different structures of product manifolds.

cs.LG↗

arXiv2023

Photophoretic Light-flyers with Germanium Coatings as Selective Absorbers

The goal of ultrathin lightweight photophoretic flyers, or light-flyers for short, is to levitate continuously in Earth's upper atmosphere using only sunlight for propulsive power. We previously reported light-flyers that levitated by utilizing differences in thermal accommodation coefficient (TAC) between the top and bottom of a thin film, made possible by coating their lower surfaces with carbon nanotubes (CNTs). Such designs, though successful, were limited due to their high thermal emissivity (>0.5), which prevented them from achieving high temperatures and resulted in their transferring relatively low amounts of momentum to the surrounding gas. To address this issue, we have developed light-flyers with undoped germanium layers that selectively absorb nearly 80% of visible light but are mostly transparent in the thermal infrared, with an average thermal emissivity of <0.1. Our experiments show that germanium-coated light-flyers could levitate at up to 43% lower light irradiances than mylar-CNT disks with identical sizes. In addition, we simulated our experiments using a combined first-principles-empirical model, allowing us to predict that our 2-cm-diameter disk-shaped germanium-coated light-flyers can levitate in the mesosphere (altitudes 68-78 km) under the natural sunlight (1.36 kW/m2). Similar ultrathin selective-absorber coatings can also be applied to three-dimensional light-flyers shaped like solar balloons, allowing them to carry significant payloads and thereby revolutionize long-term atmospheric exploration of Earth or Mars.

physics.app-ph↗

arXiv2023

Demystifying Random Number in Ethereum Smart Contract: Taxonomy, Vulnerability Identification, and Attack Detection

Recent years have witnessed explosive growth in blockchain smart contract applications. As smart contracts become increasingly popular and carry trillion dollars worth of digital assets, they become more of an appealing target for attackers, who have exploited vulnerabilities in smart contracts to cause catastrophic economic losses. Notwithstanding a proliferation of work that has been developed to detect an impressive list of vulnerabilities, the bad randomness vulnerability is overlooked by many existing tools. In this paper, we make the first attempt to provide a systematic analysis of random numbers in Ethereum smart contracts, by investigating the principles behind pseudo-random number generation and organizing them into a taxonomy. We also lucubrate various attacks against bad random numbers and group them into four categories. Furthermore, we present RNVulDet - a tool that incorporates taint analysis techniques to automatically identify bad randomness vulnerabilities and detect corresponding attack transactions. To extensively verify the effectiveness of RNVulDet, we construct three new datasets: i) 34 well-known contracts that are reported to possess bad randomness vulnerabilities, ii) 214 popular contracts that have been rigorously audited before launch and are regarded as free of bad randomness vulnerabilities, and iii) a dataset consisting of 47,668 smart contracts and 49,951 suspicious transactions. We compare RNVulDet with three state-of-the-art smart contract vulnerability detectors, and our tool significantly outperforms them. Meanwhile, RNVulDet spends 2.98s per contract on average, in most cases orders-of-magnitude faster than other tools. RNVulDet successfully reveals 44,264 attack transactions. Our implementation and datasets are released, hoping to inspire others.

cs.SE↗