arXiv subjects

Lizhao Zhang

Publications and source records attributed to Lizhao Zhang.

6 recordsLinked to original sources

arXiv2026

MO-SAE:Multi-Objective Stacked Autoencoders Optimization for Edge Anomaly Detection

Stacked AutoEncoders (SAE) have been widely adopted in edge anomaly detection scenarios. However, the resource-intensive nature of SAE can pose significant challenges for edge devices, which are typically resource-constrained and must adapt rapidly to dynamic and changing conditions. Optimizing SAE to meet the heterogeneous demands of real-world deployment scenarios, including high performance under constrained storage, low power consumption, fast inference, and efficient model updates, remains a substantial challenge. To address this, we propose an integrated optimization framework that jointly considers these critical factors to achieve balanced and adaptive system-level optimization. Specifically, we formulate SAE optimization for edge anomaly detection as a multi-objective optimization problem and propose MO-SAE (Multi-Objective Stacked AutoEncoders). The multiple objectives are addressed by integrating model clipping, multi-branch exit design, and a matrix approximation technique. In addition, a multi-objective heuristic algorithm is employed to effectively balance the competing objectives in SAE optimization. Our results demonstrate that the proposed MO-SAE delivers substantial improvements over the original approach. On the x86 architecture, it reduces storage space and power consumption by at least 50%, improves runtime efficiency by no less than 28%, and achieves an 11.8% compression rate, all while maintaining application performance. Furthermore, MO-SAE runs efficiently on edge devices with ARM architecture. Experimental results show a 15% improvement in inference speed, facilitating efficient deployment in cloud-edge collaborative anomaly detection systems.

cs.NE↗

arXiv2025

A volume correspondence between anti-de Sitter space and its boundary

Let $\mathbb{H}^{n+1}_1$ be the $(n+1)$-dimensional anti-de Sitter space (AdS), in this paper we propose to extend $\mathbb{H}^{n+1}_1$ conformally to another copy of $\mathbb{H}^{n+1}_1$ by gluing them along the boundary at infinity, and denote the resulting space by \emph{double anti-de Sitter space} $\mathbb{DH}^{n+1}_1$. We propose to introduce a volume $V_{n+1}(P)$ (possibly complex valued) on polytopes $P$ in $\mathbb{DH}^{n+1}_1$ whose facets all have non-degenerate metrics (called \emph{good} polytopes), and show that it is well defined and invariant under isometry, including the case that $P$ contains a non-trivial portion of $\partial\mathbb{H}^{n+1}_1$. For $n$ even, $V_{n+1}(P)$ is shown to be completely determined by the intersection of $P$ and $\partial\mathbb{H}^{n+1}_1$, which leads to the following important applications: it induces a new intrinsic (conformal) \emph{volume} on good polytopes in $\partial\mathbb{H}^{n+1}_1$ that is invariant under conformal transformations of $\partial\mathbb{H}^{n+1}_1$, and establishes an AdS-CFT type correspondence between the volumes on $\mathbb{DH}^{n+1}_1$ and $\partial\mathbb{H}^{n+1}_1$.

math.MG↗

arXiv2025

Simplices with fixed volumes of codimension 2 faces in a continuous deformation

For any $n$-dimensional simplex in the Euclidean space $\mathbb{R}^n$ with $n\ge 4$, it is asked that if a continuous deformation preserves the volumes of all the codimension 2 faces, then is it necessarily a \emph{rigid} motion. While the question remains open and the general belief is that the answer is affirmative, for all $n\ge 4$, we provide counterexamples to a variant of the question where $\mathbb{R}^n$ is replaced by a pseudo-Euclidean space $\mathbb{R}^{p,n-p}$ for some unspecified $p\ge 2$.

math.MG↗

arXiv2022

On the total volume of the double hyperbolic space

Let the \emph{double hyperbolic space} $\mathbb{DH}^n$, proposed in this paper as an extension of the hyperbolic space $\mathbb{H}^n$, contain a two-sheeted hyperboloid with the two sheets connected to each other along the boundary at infinity. We propose to extend the volume of convex polytopes in $\mathbb{H}^n$ to polytopes in $\mathbb{DH}^n$, where the volume is invariant under isometry but can possibly be complex valued. We show that the total volume of $\mathbb{DH}^n$ is equal to $i^n V_n(\mathbb{S}^n)$ for both even and odd dimensions, and prove a Schläfli differential formula (\SDF{}) for $\mathbb{DH}^n$. For $n$ odd, the volume of a polytope in $\mathbb{DH}^n$ is shown to be completely determined by its intersection with $\partial\mathbb{H}^n$ and induces a new intrinsic \emph{volume} on $\partial\mathbb{H}^n$ that is invariant under Möbius transformations.

math.MG↗

arXiv2019

Lifting degenerate simplices with a single volume constraint

Let $M^d$ be the spherical, Euclidean, or hyperbolic space of dimension $d\ge n+1$. Given any degenerate $(n+1)$-simplex $\mathbf{A}$ in $M^d$ with non-degenerate $n$-faces $F_i$, there is a natural partition of the set of $n$-faces into two subsets $X_1$ and $X_2$ such that $\sum_{X_1}V_n(F_i)=\sum_{X_2}V_n(F_i)$, except for a special spherical case where $X_2$ is the empty set and $\sum_{X_1}V_n(F_i)=V_n(\mathbb{S}^n)$ instead. For all cases, if the vertices vary smoothly in $M^d$ with a \emph{single} volume constraint that $\sum_{X_1}V_n(F_i)-\sum_{X_2}V_n(F_i)$ is preserved as a constant (0 or $V_n(\mathbb{S}^n)$), we prove that if a \emph{stress} invariant $c_{n-1}(α^{n-1})$ of the degenerate simplex is non-zero, then the vertices will be confined to a lower dimensional $M^n$ for any sufficiently small motion. This answers a question of the author and we also show that in the Euclidean case, $c_{n-1}(α^{n-1})=0$ is equivalent to the vertices of a \emph{dual} degenerate $(n+1)$-simplex lying on an $(n-1)$-sphere in $\mathbb{R}^n$.

math.MG↗

arXiv2018

Rigidity and volume preserving deformation on degenerate simplices

Given a degenerate $(n+1)$-simplex in a $d$-dimensional space $M^d$ (Euclidean, spherical or hyperbolic space, and $d\geq n$), for each $k$, $1\leq k\leq n$, Radon's theorem induces a partition of the set of $k$-faces into two subsets. We prove that if the vertices of the simplex vary smoothly in $M^d$ for $d=n$, and the volumes of $k$-faces in one subset are constrained only to decrease while in the other subset only to increase, then any sufficiently small motion must preserve the volumes of all $k$-faces; and this property still holds in $M^d$ for $d\geq n+1$ if an invariant $c_{k-1}(α^{k-1})$ of the degenerate simplex has the desired sign. This answers a question posed by the author, and the proof relies on an invariant $c_k(ω)$ we discovered for any $k$-stress $ω$ on a cell complex in $M^d$. We introduce a characteristic polynomial of the degenerate simplex by defining $f(x)=\sum_{i=0}^{n+1}(-1)^{i}c_i(α^i)x^{n+1-i}$, and prove that the roots of $f(x)$ are real for the Euclidean case. Some evidence suggests the same conjecture for the hyperbolic case.

math.MG↗