arXiv subjects
Publications and source records attributed to Hongwei Xu.
Let $M^n$ be a closed minimal submanifold in the unit sphere $\mathbb{S}^{n+q}$ with $n\geqslant 3$ and $q\geqslant 2$. Let $S$ be the squared length of its second fundamental form and $S_{\max}=\max_{p\in M}S(p)$. We prove that if $M$ is not totally geodesic, then \[ S_{\max}>\frac{2n}{3}+\frac{n-2}{182}. \]
A mesh of sovereign agents has no center: no shared clock, no shared model, and no coordinator to gather data or retrain. Its competence rests on each agent folding the projections its peers emit into a single internal state, online, from observations that arrive at irregular, unscheduled times, on a substrate whose weights it cannot retrain. Any one of these constraints is tractable on its own; folding optimally under all three at once is not. We ask what such a substrate must be, and prove two necessary conditions from one model of a self-evolving latent observed at irregular, exogenous times. Because the latent changes, its optimal estimator is time-varying: an adaptive timescale is necessary, and every fixed-gain filter is strictly suboptimal. And because arrivals are clock-free, the optimal estimate depends on the elapsed gap between them, which no gap-blind network recovers at any width or depth. This second condition is capacity-independent: scale cannot substitute for the missing dependence. The two conditions intersect in the continuous-time liquid class. An LSTM satisfies the first, a fixed continuous-time filter the second, and a multi-timescale liquid network both. Synthetic experiments confirm each: the network attains the timescale, and the separation is computed exactly. The characterization is necessary, not sufficient, and binds fixed-weight substrates: a network free to retrain reaches the class by other means. Proved per agent, the necessity binds every agent of a mesh, a structural condition on mesh intelligence.
We present a formal model of mesh inference: how a population of independent agents, each holding private state and exchanging only admitted, typed observations, derives a conclusion none of them holds alone, with no central coordinator and no agent exposed. No agent shares weights, gradients, or hidden state, and the agents may span different teams, networks, and organizations. Motivated by the observation that asking a model is energy-minimizing inference, we model the mesh as a coupled free energy that each agent relaxes locally. We show that a single admission/emission policy governs three properties. First, mesh inference converges to a unique answer for any admission, symmetric or not, because the coupling is always an M-matrix. Second, it is identification-complete: it derives the centralized optimum exactly when the contributing views are carrier-connected. Third, it is observation-only: no node transmits its internals, and confidentiality is the dual of identification. Content-addressed lineage is the only global side-channel. In the linear-Gaussian regime every derived answer is determined, hence equal to the centralized optimum, at O(diam^2) latency, the measured price of removing the center. One such derivation is one turn of a center-free learning loop, which we formalize as architecture rather than prove. The open problem we state is when asking improves the collective rather than corrupting it: whether the non-linear closure derives an upgraded answer or a confident error. To our knowledge, this is the first formal characterization of when a center-free, observation-only mesh recovers the centralized optimum.
Teams of LLM agents increasingly collaborate on tasks spanning days or weeks: multi-day data-generation sprints where generator, reviewer, and auditor agents coordinate in real time on overlapping batches; specialists carrying findings forward across session restarts; product decisions compounding over many review rounds. This requires agents to share, evaluate, and combine each other's cognitive state in real time across sessions. We call this cross-session agent-to-agent cognitive collaboration, distinct from parallel agent execution. To enable it, three problems must be solved together. (P1) Each agent decides field by field what to accept from peers, not accept or reject whole messages. (P2) Every claim is traceable to source, so returning claims are recognised as echoes of the receiver's own prior thinking. (P3) Memory that survives session restarts is relevant because of how it was stored, not how it is retrieved. These are protocol-level properties at the semantic layer of agent communication, distinct from tool-access and task-delegation protocols at lower layers. We call this missing protocol layer "semantic infrastructure," and the Mesh Memory Protocol (MMP) specifies it. Four composable primitives work together: CAT7, a fixed seven-field schema for every Cognitive Memory Block (CMB); SVAF, which evaluates each field against the receiver's role-indexed anchors and realises P1; inter-agent lineage, carried as parents and ancestors of content-hash keys and realising P2; and remix, which stores only the receiver's own role-evaluated understanding of each accepted CMB, never the raw peer signal, realising P3. MMP is specified, shipped, and running in production across three reference deployments, where each session runs an autonomous agent as a mesh peer with its own identity and memory, collaborating with other agents across the network for collective intelligence.
A notable difference between the ordinary and Hadamard products is that the Hadamard product of two singular positive semidefinite matrices can be nonsingular, and one of the factors can even be indefinite. We present an eigenvalue lower bound for a Hadamard product that depends on the rank, effective condition number, and diagonal entries of one factor, and the smallest eigenvalues of certain principal submatrices of the other factor. We give numerical examples and discuss its applications in array signal processing and matrix time series analysis.
MeloTune is an iPhone-deployed music agent that instantiates the Mesh Memory Protocol (MMP) and Symbolic-Vector Attention Fusion (SVAF) as a production system for affect-aware music curation with peer-to-peer mood coupling. Each device runs two closed-form continuous-time (CfC) networks: a private listener-level CfC that predicts a short-horizon affective trajectory on Russell's circumplex and drives proactive curation, and a shared mesh-runtime CfC at MMP Layer 6 that integrates Cognitive Memory Blocks (CMBs) from co-listening peers. CfC hidden states never cross the wire; only structured CMBs do. A Personal Arousal Function (PAF) replaces the standard linear mapping from audio intensity to psychological arousal with a per-listener learned adjustment, trained from behavioral signals (skip, completion, favorite, volume) and from drift between user-declared mood and machine inference. The same track receives different arousal predictions for different listeners. The model (94,552 parameters) achieves trajectory MAE 0.414, pattern accuracy 96.6%, and intent accuracy 69.4% on held-out validation. PAF evidence from a live deployment session (46 observations across 11 genres) demonstrates that the learning loop operates end-to-end, with pop reaching full confidence after 22 observations. All inference runs on-device via CoreML. To our knowledge, this is the first production deployment of MMP/SVAF on consumer mobile hardware. The accompanying SDK (sym-swift v0.3.78, SYMCore v0.3.7) enforces strict protocol conformance. Music is the case study; the substrate is the contribution.
When autonomous agents observe different domains of a shared environment, each signal they exchange mixes relevant and irrelevant dimensions. No existing mechanism lets the receiver evaluate which dimensions to absorb. We introduce Symbolic-Vector Attention Fusion (SVAF), the content-evaluation half of a two-level coupling engine for collective intelligence. SVAF decomposes each inter-agent signal into 7 typed semantic fields, evaluates each through a learned fusion gate, and produces a remix -- new knowledge from the intersection of two domains. A band-pass model yields four outcomes (redundant, aligned, guarded, rejected), solving both selectivity and redundancy. The fusion gate independently discovers a cross-domain relevance hierarchy: mood emerges as the highest-weight field by epoch 1, before accuracy plateaus -- consistent with independent mechanistic evidence that LLM emotion representations are structurally embedded along valence-arousal axes. SVAF forms Layer 4 of the Mesh Memory Protocol (MMP); the other half of the coupling engine is a per-agent Closed-form Continuous-time (CfC) neural network at Layer 6, whose learned per-neuron time constants (tau) create the temporal dynamics from which collective intelligence emerges: fast neurons synchronise affect across agents in seconds, while slow neurons preserve domain expertise indefinitely. SVAF determines what enters each agent's cognitive state; CfC determines how that state evolves. Trained on 237K samples from 273 narrative scenarios, SVAF achieves 78.7% three-class accuracy. We verify the complete mesh cognition loop -- from per-field evaluation through remix, CfC state evolution, tau-modulated peer blending, and autonomous action -- in a live deployment with 7 nodes across macOS, iOS, and web.
In this paper, we prove that any closed minimal hypersurface $M^4$ in the $5$-dimensional unit sphere $\mathbb{S}^5$ with constant scalar curvature and constant $3$-th mean curvature must be isoparametric. To be precise, $M^4$ is either an equatorial 4-sphere, a product of spheres $\mathbb{S} ^{2}(\frac{\sqrt{2}}{2}) \times \mathbb{S} ^{2}(\frac{\sqrt{2}}{2})$ or $\mathbb{S} ^{1}(\frac{1}{2}) \times \mathbb{S} ^{3}(\frac{\sqrt{3}}{2})$, or a Cartan's minimal hypersurface. In particular, the value of the squared norm of the second fundamental form $S$ can only be 0, 4, or 12. This result strongly supports Chern's conjecture.
This paper addresses the problem of estimating the Hermitian Toeplitz covariance matrix under practical hardware constraints of sparse observations and coarse quantization. Within the triangular-dithered quantization framework, we propose an estimator called Toeplitz-projected sample covariance matrix (Q-TSCM) to compensate for the quantization-induced bias, together with its finite-bit counterpart termed the $2k$-bit Toeplitz-projected sample covariance matrix ($2k$-TSCM), obtained by truncating the pre-quantization observations. Under the complex Gaussian assumption, we derive non-asymptotic error bounds of the estimators that reveal a quadratic dependence on the quantization level and capture the effect of sparse sampling patterns through the so-called coverage coefficient. To further improve performance, we propose the quantized sparse and parametric approach (Q-SPA) based on a covariance-fitting criterion, which enforces additionally positive semidefiniteness at the cost of solving a semidefinite program. Numerical experiments are presented that corroborate our theoretical findings and demonstrate the effectiveness of the proposed estimators in the application to direction-of-arrival estimation.
We investigate the lower bound for higher eigenvalues $\lambda_i$ of the poly-Laplace operator on a bounded domain and improve the famous Li-Yau inequality and its related results. Firstly, we consider the low dimensional cases for the P\'{o}lya conjecture, the clamped plate problem and the eigenvalue problem of the poly-Laplacian and deliver a series of deep eigenvalue inequalities for these problems respectively. Secondly, we establish a sharp lower bound for the eigenvalues of the poly-Laplacia in arbitrary dimension under some certain restrictive conditions. Finally, we provide an improved inequality for $\lambda_i$ in arbitrary dimension without any restrictive conditions. Our results also yield the improvement of the lower bounds for the Stokes eigenvalue problems and the Generalized P\'{o}lya conjecture.
This paper addresses the challenge of Toeplitz covariance matrix estimation from partial entries of random quantized samples. To balance trade-offs among the number of samples, the number of entries observed per sample, and the data resolution, we propose a ruler-based quantized Toeplitz covariance estimator. We derive non-asymptotic error bounds and analyze the convergence rates of the proposed estimator. Our results show that the estimator is near-optimal and imply that reducing data resolution within a certain range has a limited impact on the estimation accuracy. Numerical experiments are provided that validate our theoretical findings and show effectiveness of the proposed estimator.
Audio-Driven Face Animation is an eagerly anticipated technique for applications such as VR/AR, games, and movie making. With the rapid development of 3D engines, there is an increasing demand for driving 3D faces with audio. However, currently available 3D face animation datasets are either scale-limited or quality-unsatisfied, which hampers further developments of audio-driven 3D face animation. To address this challenge, we propose MMFace4D, a large-scale multi-modal 4D (3D sequence) face dataset consisting of 431 identities, 35,904 sequences, and 3.9 million frames. MMFace4D exhibits two compelling characteristics: 1) a remarkably diverse set of subjects and corpus, encompassing actors spanning ages 15 to 68, and recorded sentences with durations ranging from 0.7 to 11.4 seconds. 2) It features synchronized audio and 3D mesh sequences with high-resolution face details. To capture the subtle nuances of 3D facial expressions, we leverage three synchronized RGBD cameras during the recording process. Upon MMFace4D, we construct a non-autoregressive framework for audio-driven 3D face animation. Our framework considers the regional and composite natures of facial animations, and surpasses contemporary state-of-the-art approaches both qualitatively and quantitatively. The code, model, and dataset will be publicly available.
The softmax-based loss functions and its variants (e.g., cosface, sphereface, and arcface) significantly improve the face recognition performance in wild unconstrained scenes. A common practice of these algorithms is to perform optimizations on the multiplication between the embedding features and the linear transformation matrix. However in most cases, the dimension of embedding features is given based on traditional design experience, and there is less-studied on improving performance using the feature itself when giving a fixed size. To address this challenge, this paper presents a softmax approximation method called SubFace, which employs the subspace feature to promote the performance of face recognition. Specifically, we dynamically select the non-overlapping subspace features in each batch during training, and then use the subspace features to approximate full-feature among softmax-based loss, so the discriminability of the deep model can be significantly enhanced for face recognition. Comprehensive experiments conducted on benchmark datasets demonstrate that our method can significantly improve the performance of vanilla CNN baseline, which strongly proves the effectiveness of subspace strategy with the margin-based loss.
We propose a real time deep learning framework for video-based facial expression capture. Our process uses a high-end facial capture pipeline based on FACEGOOD to capture facial expression. We train a convolutional neural network to produce high-quality continuous blendshape weight output from video training. Since this facial capture is fully automated, our system can drastically reduce the amount of labor involved in the development of modern narrative-driven video games or films involving realistic digital doubles of actors and potentially hours of animated dialogue per character. We demonstrate compelling animation inference in challenging areas such as eyes and lips.
In this paper, we investigate Liu-Xu-Ye-Zhao's conjecture [30] and prove a sharp convergence theorem for the mean curvature flow of arbitrary codimension in spheres which improves the convergence theorem of Baker [2] as well as the differentiable sphere theorems of Gu-Xu-Zhao [16, 50, 52].
In this paper, we give some lower bounds for several eigenvalues. Firstly, we investigate the eigenvalues $λ_i$ of the Laplace operator and prove a sharp lower bound. Moreover, we extent this estimate of the eigenvalues to general cases. Secondly, we study the eigenvalues $Γ_i$ for the clamped plate problem and deliver a sharp bound for the clamped plate problem for arbitrary dimension.
For a set $A$, let $P(A)$ be the set of all finite subset sums of $A$. In this paper, for a sequence of integers $B=\{1<b_1<b_2<\cdots\}$ and $3b_1+5\leq b_2\leq 6b_1+10$, we determine the critical value for $b_3$ such that there exists an infinite sequence $A$ of positive integers for which $P(A)=\mathbb{N}\setminus B$. This result shows that we partially solve the problem of Fang and Fang [`On an inverse problem in additive number theory', Acta Math. Hungar. 158(2019), 36-39].
For a set $A$, let $P(A)$ be the set of all finite subset sums of $A$. We prove that if a sequence $B=\{11\leq b_1<b_2<\cdots\}$ satisfies $b_2=3b_1+5$, $b_3=3b_2+2$ and $b_{n+1}=3b_n+4b_{n-1}$ for all $n\geq 3$, then there is a sequence of positive integers $A=\{a_1<a_2<\cdots\}$ such that $P(A)=\mathbb{N}\setminus B$. This result shows that the answer to the problem of Chen and Wu [`The inverse problem on subset sums', European. J. Combin. 34(2013), 841-845] is negative.