arXiv subjects

Yifeng Huang

Publications and source records attributed to Yifeng Huang.

At least 37 records · Page 2Linked to original sources

arXiv2024

Commuting matrices via commuting endomorphisms

Evidences have suggested that counting representations are sometimes tractable even when the corresponding classification problem is almost impossible, or "wild" in a precise sense. Such counting problems are directly related to matrix counting problems, many of which are under active research. Using a general framework we formulate for such counting problems, we reduce some counting problems about commuting matries to problems about endomorphisms on all finite abelian $p$-groups. As an application, we count finite modules on some first examples of nonreduced curves over $\mathbb{F}_q$. We also relate some classical and hard problems regarding commuting triples of matrices to a conjecture of Onn on counting conjugacy classes of the automorphism group of an arbitrary finite abelian $p$-group.

math.RT↗

arXiv2024

Cokernels of random matrix products and flag Cohen--Lenstra heuristic

In [NVP22], Nguyen and Van Peski raised the question of whether the surjective flag of $\mathbb Z_p$-modules modeled by $\mathrm{cok}(M_1\cdots M_k)\twoheadrightarrow \dots\twoheadrightarrow \mathrm{cok}(M_1)$ for independent random matrices $M_1,\dots,M_k\in \mathrm{Mat}_n(\mathbb Z_p)$ satisfies the Cohen--Lenstra heuristic. We answer the question affirmatively when $M_1,\dots,M_k$ follow the Haar measure, and our proof demonstrates how classical ideas in Cohen--Lenstra heuristic adapt naturally to the flag setting. We also prove an analogue for non-square matrices.

math.NT↗

arXiv2024

Inelastic electron scattering at large angles: the phonon polariton contribution

We explore the inelastic electron scattering in SrTiO3, PbTiO3, and SiC in their phonon energy range, challenging the assumption that phonon polaritons are excluded at large angles in high-resolution transmission electron energy-loss spectroscopy. We demonstrate that through multiple scattering, the electron beam can excite both phonons and phonon polaritons, and the relative proportion of each varies depending on the structure factor and scattering angle. Integrating dielectric theory, density functional theory, and multi-slice simulations, we provide a comprehensive framework for understanding these interactions in materials with polar optical phonons.

cond-mat.mtrl-sci↗

arXiv2023

Count What You Want: Exemplar Identification and Few-shot Counting of Human Actions in the Wild

This paper addresses the task of counting human actions of interest using sensor data from wearable devices. We propose a novel exemplar-based framework, allowing users to provide exemplars of the actions they want to count by vocalizing predefined sounds ''one'', ''two'', and ''three''. Our method first localizes temporal positions of these utterances from the audio sequence. These positions serve as the basis for identifying exemplars representing the action class of interest. A similarity map is then computed between the exemplars and the entire sensor data sequence, which is further fed into a density estimation module to generate a sequence of estimated density values. Summing these density values provides the final count. To develop and evaluate our approach, we introduce a diverse and realistic dataset consisting of real-world data from 37 subjects and 50 action categories, encompassing both sensor and audio data. The experiments on this dataset demonstrate the viability of the proposed method in counting instances of actions from new classes and subjects that were not part of the training data. On average, the discrepancy between the predicted count and the ground truth value is 7.47, significantly lower than the errors of the frequency-based and transformer-based methods. Our project, code and dataset can be found at https://github.com/cvlab-stonybrook/ExRAC.

cs.CV↗

arXiv2023

Real-Space Visualization of Frequency-Dependent Anisotropy of Atomic Vibrations

The underlying dielectric properties of materials, intertwined with intriguing phenomena such as topological polariton modes and anisotropic thermal conductivities, stem from the anisotropy in atomic vibrations. Conventionally, X-ray diffraction techniques have been employed to estimate thermal ellipsoids of distinct elements, albeit lacking the desired spatial and energy resolutions. Here we introduce a novel approach utilizing the dark-field monochromated electron energy-loss spectroscopy for momentum-selective vibrational spectroscopy, enabling the cartographic delineation of variations of phonon polarization vectors. By applying this technique to centrosymmetric cubic-phase strontium titanate, we successfully discern two types of oxygen atoms exhibiting contrasting vibrational anisotropies below and above 60 meV due to their frequency-linked thermal ellipsoids. This method establishes a new pathway to visualize phonon eigenvectors at specific crystalline sites for diverse elements, thus delving into uncharted realms of dielectric, optical, and thermal property investigations with unprecedented spatial resolutions.

cond-mat.mtrl-sci↗

arXiv2023

The cokernel of a polynomial push-forward of a random integral matrix with concentrated residue

We prove new statistical results about the distribution of the cokernel of a random integral matrix with a concentrated residue. Given a prime $p$ and a positive integer $n$, consider a random $n \times n$ matrix $X_n$ over the ring $\mathbb{Z}_p$ of $p$-adic integers whose entries are independent. Previously, Wood showed that regardless of the distribution of $X_n$, as long as each entry of $X_n$ is not too concentrated on a single residue modulo $p$, the distribution of the cokernel $\mathrm{cok}(X_n)$ of $X_n$, up to isomorphism, weakly converges to the Cohen--Lenstra distribution, as $n \rightarrow \infty$. In this paper, we consider the case when $X_n$ has a concentrated residue $A_n$ so that $X_n = A_n + pB_n$, where $B_n$ is a random $n \times n$ matrix over $\mathbb{Z}_p$. We show that for every fixed $n$ and a non-constant monic polynomial $P(t) \in \mathbb{Z}_p[t]$, we can explicitly compute the distribution of $\mathrm{cok}(P(X_n))$ when $B_n$ is a Haar-random matrix. Using this, we also show that for specific choices of $A_n$ a much wider class of random matrices $B_n$ gives the same distribution of $\mathrm{cok}(P(X_n))$. For the Haar-random $B_n$, we deduce our result from an interesting equidistribution result for matrices over $\mathbb{Z}_p[t]/(P(t))$, which we prove by establishing a version of the Weierstrass preparation theorem for the noncommutative ring $\mathrm{M}_n(\mathbb{Z}_p)$ of $n \times n$ matrices over $\mathbb{Z}_p$.

math.NT↗

arXiv2023

Interactive Class-Agnostic Object Counting

We propose a novel framework for interactive class-agnostic object counting, where a human user can interactively provide feedback to improve the accuracy of a counter. Our framework consists of two main components: a user-friendly visualizer to gather feedback and an efficient mechanism to incorporate it. In each iteration, we produce a density map to show the current prediction result, and we segment it into non-overlapping regions with an easily verifiable number of objects. The user can provide feedback by selecting a region with obvious counting errors and specifying the range for the estimated number of objects within it. To improve the counting result, we develop a novel adaptation loss to force the visual counter to output the predicted count within the user-specified range. For effective and efficient adaptation, we propose a refinement module that can be used with any density-based visual counter, and only the parameters in the refinement module will be updated during adaptation. Our experiments on two challenging class-agnostic object counting benchmarks, FSCD-LVIS and FSC-147, show that our method can reduce the mean absolute error of multiple state-of-the-art visual counters by roughly 30% to 40% with minimal user input. Our project can be found at https://yifehuang97.github.io/ICACountProjectPage/.

cs.CV↗

arXiv2023

Punctual Quot schemes and Cohen--Lenstra series of the cusp singularity

The Quot scheme of points $\mathrm{Quot}_{d,n}(X)$ on a variety $X$ over a field $k$ parametrizes quotient sheaves of $\mathcal{O}_X^{\oplus d}$ of zero-dimensional support and length $n$. It is a rank-$d$ generalization of the Hilbert scheme of $n$ points. When $X$ is a reduced curve with only the cusp singularity $\{x^2=y^3\}$ and $d\geq 0$ is fixed, the generating series for the motives of $\mathrm{Quot}_{d,n}(X)$ in the Grothendieck ring of varieties is studied via Gröbner bases, and shown to be rational. Moreover, the generating series is computed explicitly when $d\leq 3$. The computational results exhibit surprising patterns (despite the fact that the category of finite length coherent modules over a cusp is wild), which not only enable us to conjecture the exact form of the generating series for all $d$, but also suggest a general functional equation whose $d=1$ case is the classical functional equation of the motivic zeta function known for any Gorenstein curve. As another side of the story, Quot schemes are related to the Cohen--Lenstra series. The Cohen--Lenstra series encodes the count of "commuting matrix points'' (or equivalently, coherent modules of finite length) of a variety over a finite field, about which Huang formuated a "rationality'' conjecture for singular curves. We prove a general formula that expresses the Cohen--Lenstra series in terms of the motives of the (punctual) Quot schemes, which together with our main rationality theorem, provides positive evidence for Huang's conjecture for the cusp.

math.AG↗

arXiv2023

Counting matrix points on certain varieties over finite fields

Classical hypergeometric functions are well-known to play an important role in arithmetic algebraic geometry. These functions offer solutions to ordinary differential equations, and special cases of such solutions are periods of Picard-Fuchs varieties of Calabi-Yau type. Gauss' $_2F_1$ includes the celebrated case of elliptic curves through the theory of elliptic functions. In the 80s, Greene defined finite field hypergeometric functions that can be used to enumerate the number of finite field points on such varieties. We extend some of these results to count finite field ``matrix points." For example, for every $n\geq 1,$ we consider the matrix elliptic curves $$ B^2 = A(A-I_n)(A-a I_n), $$ where $(A,B)$ are commuting $n\times n$ matrices over a finite field $\mathbb{F}_q$ and $a\neq 0,1$ is fixed. Our formulas are assembled from Greene's hypergeometric functions and $q$-multinomial coefficients. We use these formulas to prove Sato-Tate distributions for the error terms for matrix point counts for these curves and some families of $K3$ surfaces.

math.NT↗

arXiv2022

Machine-Learning Compression for Particle Physics Discoveries

In collider-based particle and nuclear physics experiments, data are produced at such extreme rates that only a subset can be recorded for later analysis. Typically, algorithms select individual collision events for preservation and store the complete experimental response. A relatively new alternative strategy is to additionally save a partial record for a larger subset of events, allowing for later specific analysis of a larger fraction of events. We propose a strategy that bridges these paradigms by compressing entire events for generic offline analysis but at a lower fidelity. An optimal-transport-based $β$ Variational Autoencoder (VAE) is used to automate the compression and the hyperparameter $β$ controls the compression fidelity. We introduce a new approach for multi-objective learning functions by simultaneously learning a VAE appropriate for all values of $β$ through parameterization. We present an example use case, a di-muon resonance search at the Large Hadron Collider (LHC), where we show that simulated data compressed by our $β$-VAE has enough fidelity to distinguish distinct signal morphologies.

hep-ph↗

arXiv2022

Mutually annihilating matrices, and a Cohen--Lenstra series for the nodal singularity

We give a generating function for the number of pairs of $n\times n$ matrices $(A, B)$ over a finite field that are mutually annihilating, namely, $AB=BA=0$. This generating function can be viewed as a singular analogue of a series considered by Cohen and Lenstra. We show that this generating function has a factorization that allows it to be meromorphically extended to the entire complex plane. We also use it to count pairs of mutually annihilating nilpotent matrices. This work is essentially a study of the motivic aspects about the variety of modules over $\mathbb C[u,v]/(uv)$ as well as the moduli stack of coherent sheaves over an algebraic curve with nodal singularities.

math.AG↗

arXiv2022

The Devil is in the Labels: Noisy Label Correction for Robust Scene Graph Generation

Unbiased SGG has achieved significant progress over recent years. However, almost all existing SGG models have overlooked the ground-truth annotation qualities of prevailing SGG datasets, i.e., they always assume: 1) all the manually annotated positive samples are equally correct; 2) all the un-annotated negative samples are absolutely background. In this paper, we argue that both assumptions are inapplicable to SGG: there are numerous "noisy" groundtruth predicate labels that break these two assumptions, and these noisy samples actually harm the training of unbiased SGG models. To this end, we propose a novel model-agnostic NoIsy label CorrEction strategy for SGG: NICE. NICE can not only detect noisy samples but also reassign more high-quality predicate labels to them. After the NICE training, we can obtain a cleaner version of SGG dataset for model training. Specifically, NICE consists of three components: negative Noisy Sample Detection (Neg-NSD), positive NSD (Pos-NSD), and Noisy Sample Correction (NSC). Firstly, in Neg-NSD, we formulate this task as an out-of-distribution detection problem, and assign pseudo labels to all detected noisy negative samples. Then, in Pos-NSD, we use a clustering-based algorithm to divide all positive samples into multiple sets, and treat the samples in the noisiest set as noisy positive samples. Lastly, in NSC, we use a simple but effective weighted KNN to reassign new predicate labels to noisy positive samples. Extensive results on different backbones and tasks have attested to the effectiveness and generalization abilities of each component of NICE.

cs.CV↗

arXiv2022

From the Greene--Wu Convolution to Gradient Estimation over Riemannian Manifolds

Over a complete Riemannian manifold of finite dimension, Greene and Wu introduced a convolution, known as Greene-Wu (GW) convolution. In this paper, we study properties of the GW convolution and apply it to non-Euclidean machine learning problems. In particular, we derive a new formula for how the curvature of the space would affect the curvature of the function through the GW convolution. Also, following the study of the GW convolution, a new method for gradient estimation over Riemannian manifolds is introduced.

cs.LG↗

arXiv2021

Counting on the variety of modules over the quantum plane

Let $ζ$ be a fixed nonzero element in a finite field $\mathbb F_q$ with $q$ elements. In this article, we count the number of pairs $(A,B)$ of $n\times n$ matrices over $\mathbb F_q$ satisfying $AB=ζBA$ by giving a generating function. This generalizes a generating function of Feit and Fine that counts pairs of commuting matrices. Our result can be also viewed as the point count of the variety of modules over the quantum plane $xy=ζyx$, whose geometry was described by Chen and Lu.

math.AG↗

arXiv2021

Continuously Increasing Subsequences of Random Multiset Permutations

For a word $π$ and integer $i$, we define $L^i(π)$ to be the length of the longest subsequence of the form $i(i+1)\cdots j$, and we let $L(π):=\max_i L^i(π)$. In this paper we estimate the expected values of $L^1(π)$ and $L(π)$ when $π$ is chosen uniformly at random from all words which use each of the first $n$ integers exactly $m$ times. We show that $\mathbb{E}[L^1(π)]\sim m$ if $n$ is sufficiently larger in terms of $m$ as $m$ tends towards infinity, confirming a conjecture of Diaconis, Graham, He, and Spiro. We also show that $\mathbb{E}[L(π)]$ is asymptotic to the inverse gamma function $Γ^{-1}(n)$ if $n$ is sufficiently large in terms of $m$ as $m$ tends towards infinity.

math.CO↗

arXiv2021

Betti and Hodge numbers of configuration spaces of a punctured elliptic curve from its zeta functions

Given an elliptic curve $E$ defined over $\mathbb{C}$, let $E^{\times}$ be an open subset of $E$ obtained by removing a point. In this paper, we show that the $i$-th Betti number of the unordered configuration space $\mathrm{Conf}^{n}(E^{\times})$ of $n$ points on $E^{\times}$ appears as a coefficient of an explicit rational function in two variables. We also compute its Hodge numbers as coefficients of another explicit rational function in four variables. Our result is interesting because these rational functions resemble the generating function of the $\mathbb{F}_{q}$-point counts of $\mathrm{Conf}^{n}(E^{\times})$, which can be obtained from the zeta function of $E$ over a finite field $\mathbb{F}_{q}$. We show that the mixed Hodge structure of the $i$-th singular cohomology group $H^{i}(\mathrm{Conf}^{n}(E^{\times}))$ with complex coefficients is pure of weight $w(i)$, an explicit integer we provide in this paper. This purity statement implies our main result about the Betti numbers and the Hodge numbers. Our proof uses Totaro's spectral sequence computation that describes the weight filtration of the mixed Hodge structure on $H^{i}(\mathrm{Conf}^{n}(E^{\times}))$.

math.AG↗

arXiv2021

Video Relation Detection via Tracklet based Visual Transformer

Video Visual Relation Detection (VidVRD), has received significant attention of our community over recent years. In this paper, we apply the state-of-the-art video object tracklet detection pipeline MEGA and deepSORT to generate tracklet proposals. Then we perform VidVRD in a tracklet-based manner without any pre-cutting operations. Specifically, we design a tracklet-based visual Transformer. It contains a temporal-aware decoder which performs feature interactions between the tracklets and learnable predicate query embeddings, and finally predicts the relations. Experimental results strongly demonstrate the superiority of our method, which outperforms other methods by a large margin on the Video Relation Understanding (VRU) Grand Challenge in ACM Multimedia 2021. Codes are released at https://github.com/Dawn-LX/VidVRD-tracklets.

cs.CV↗

arXiv2020

Unit equations on quaternions

A classical result about unit equations says that if $Γ_1$ and $Γ_2$ are finitely generated subgroups of $\mathbb C^\times$, then the equation $x+y=1$ has only finitely many solutions with $x\inΓ_1$ and $y\in Γ_2$. We study a noncommutative analogue of the result, where $Γ_1,Γ_2$ are finitely generated subsemigroups of the multiplicative group of a quaternion algebra. We prove an analogous conclusion when both semigroups are generated by algebraic quaternions with norms greater than 1 and one of the semigroups is commutative. As an application in dynamics, we prove that if $f$ and $g$ are endomorphisms of a curve $C$ of genus $1$ over an algebraically closed field $k$, and $\mathrm{deg}(f), \mathrm{deg}(g)\geq 2$, then $f$ and $g$ have a common iterate if and only if some forward orbit of $f$ on $C(k)$ has infinite intersection with an orbit of $g$.

math.NT↗