arXiv subjects

Dingkang Wang

Publications and source records attributed to Dingkang Wang.

At least 19 recordsLinked to original sources

arXiv2026

Matrix equivalence to Smith normal form: new theoretical results for multivariate polynomial matrices

This paper investigates the Smith normal form equivalence problem for multivariate polynomial matrices. Using methods from matrix theory and polynomial ideal theory, we prove that Frost and Storey's 1978 conjecture holds for a broad class of matrices: such a matrix is equivalent to its Smith normal form if and only if its reduced minors of each order generate the unit ideal. Moreover, by extending the original matrix class via automorphisms of the polynomial ring, we show that our framework applies in a substantially more general setting.

math.AC↗

arXiv2026

Formalizing Wu-Ritt Method in Lean 4

We formalize the Wu-Ritt characteristic set method for the triangular decomposition of polynomial systems in the Lean 4 theorem prover. Our development includes the core algebraic notions of the method, such as polynomial initials, orders, pseudo-division, pseudo-remainders with respect to a polynomial or a triangular set, and standard and weak ascending sets. On this basis, we formalize algorithms for computing basic sets, characteristic sets, and zero decompositions, and prove their termination and correctness. In particular, we formalize the well-ordering principle relating a polynomial system to its characteristic set and verify that zero decomposition expresses the zero set of the original system as a union of zero sets of triangular sets away from the zeros of the corresponding initials. This work provides a machine-checked verification of Wu-Ritt's method in Lean 4 and establishes a foundation for certified polynomial system solving and geometric theorem proving.

math.AC↗

arXiv2025

Smith normal forms of bivariate polynomial matrices

In 1978, Frost and Storey asserted that a bivariate polynomial matrix is equivalent to its Smith normal form if and only if the reduced minors of all orders generate the unit ideal. In this paper, we first demonstrate by constructing an example that for any given positive integer s with s >= 2, there exists a square bivariate polynomial matrix M with the degree of det(M) in y equal to s, for which the condition that reduced minors of all orders generate the unit ideal is not a sufficient condition for M to be equivalent to its Smith normal form. Subsequently, we prove that for any square bivariate polynomial matrix M where the degree of det(M) in y is at most 1, Frost and Storey's assertion holds. Using the Quillen-Suslin theorem, we further extend our consideration of M to rank-deficient and non-square cases.

cs.SC↗

arXiv2025

Movie Gen: A Cast of Media Foundation Models

We present Movie Gen, a cast of foundation models that generates high-quality, 1080p HD videos with different aspect ratios and synchronized audio. We also show additional capabilities such as precise instruction-based video editing and generation of personalized videos based on a user's image. Our models set a new state-of-the-art on multiple tasks: text-to-video synthesis, video personalization, video editing, video-to-audio generation, and text-to-audio generation. Our largest video generation model is a 30B parameter transformer trained with a maximum context length of 73K video tokens, corresponding to a generated video of 16 seconds at 16 frames-per-second. We show multiple technical innovations and simplifications on the architecture, latent spaces, training objectives and recipes, data curation, evaluation protocols, parallelization techniques, and inference optimizations that allow us to reap the benefits of scaling pre-training data, model size, and training compute for training large scale media generation models. We hope this paper helps the research community to accelerate progress and innovation in media generation models. All videos from this paper are available at https://go.fb.me/MovieGenResearchVideos.

cs.CV↗

arXiv2024

Two Algorithms for Computing Rational Univariate Representations of Zero-Dimensional Ideals with Parameters

Based on the partition of parameter space, two algorithms for computing the rational univariate representation of zero-dimensional ideals with parameters are presented in the paper. Unlike the rational univariate representation of zero-dimensional ideals without parameters, the number of zeros of zero-dimensional ideals with parameters under various specializations is different, which leads to choosing and checking the separating element, the key to computing the rational univariate representation, is difficult. In order to pick out the separating element, we first ensure that under each branch the ideal has the same number of zeros by partitioning the parameter space. Subsequently two ideas are given to choose and check the separating element. One idea is that by extending the subresultant theorem to parametric cases, we utilize the extended subresultant theorem to choose the separating element with the further partition of parameter space and then with the help of parametric greatest common divisor theory compute rational univariate representations. Another one is that we go straight to choose and check the separating element by the computation of parametric greatest common divisors, then immediately get the rational univariate representations. Based on these, we design two different algorithms for computing rational univariate representations of zero-dimensional ideals with parameters. Furthermore, the algorithms have been implemented on Singular and the performance comparison are presented.

cs.SC↗

arXiv2024

On the equivalence problem of Smith forms for multivariate polynomial matrices

This paper delves into the equivalence problem of Smith forms for multivariate polynomial matrices. Generally speaking, multivariate ($n \geq 2$) polynomial matrices and their Smith forms may not be equivalent. However, under certain specific condition, we derive the necessary and sufficient condition for their equivalence. Let $F\in K[x_1,\ldots,x_n]^{l\times m}$ be of rank $r$, $d_r(F)\in K[x_1]$ be the greatest common divisor of all the $r\times r$ minors of $F$, where $K$ is a field, $x_1,\ldots,x_n$ are variables and $1 \leq r \leq \min\{l,m\}$. Our key findings reveal the result: $F$ is equivalent to its Smith form if and only if all the $i\times i$ reduced minors of $F$ generate $K[x_1,\ldots,x_n]$ for $i=1,\ldots,r$.

cs.SC↗

arXiv2024

Design of an Adaptive Lightweight LiDAR to Decouple Robot-Camera Geometry

A fundamental challenge in robot perception is the coupling of the sensor pose and robot pose. This has led to research in active vision where robot pose is changed to reorient the sensor to areas of interest for perception. Further, egomotion such as jitter, and external effects such as wind and others affect perception requiring additional effort in software such as image stabilization. This effect is particularly pronounced in micro-air vehicles and micro-robots who typically are lighter and subject to larger jitter but do not have the computational capability to perform stabilization in real-time. We present a novel microelectromechanical (MEMS) mirror LiDAR system to change the field of view of the LiDAR independent of the robot motion. Our design has the potential for use on small, low-power systems where the expensive components of the LiDAR can be placed external to the small robot. We show the utility of our approach in simulation and on prototype hardware mounted on a UAV. We believe that this LiDAR and its compact movable scanning design provide mechanisms to decouple robot and sensor geometry allowing us to simplify robot perception. We also demonstrate examples of motion compensation using IMU and external odometry feedback in hardware.

cs.RO↗

arXiv2024

Animated Stickers: Bringing Stickers to Life with Video Diffusion

We introduce animated stickers, a video diffusion model which generates an animation conditioned on a text prompt and static sticker image. Our model is built on top of the state-of-the-art Emu text-to-image model, with the addition of temporal layers to model motion. Due to the domain gap, i.e. differences in visual and motion style, a model which performed well on generating natural videos can no longer generate vivid videos when applied to stickers. To bridge this gap, we employ a two-stage finetuning pipeline: first with weakly in-domain data, followed by human-in-the-loop (HITL) strategy which we term ensemble-of-teachers. It distills the best qualities of multiple teachers into a smaller student model. We show that this strategy allows us to specifically target improvements to motion quality while maintaining the style from the static image. With inference optimizations, our model is able to generate an eight-frame video with high-quality, interesting, and relevant motion in under one second.

cs.CV↗

arXiv2021

Ordinally Consensus Subset over Multiple Metrics

In this paper, we propose to study the following maximum ordinal consensus problem: Suppose we are given a metric system (M, X), which contains k metrics M = {ρ_1,..., ρ_k} defined on the same point set X. We aim to find a maximum subset X' of X such that all metrics in M are "consistent" when restricted on the subset X'. In particular, our definition of consistency will rely only on the ordering between pairwise distances, and thus we call a "consistent" subset an ordinal consensus of X w.r.t. M. We will introduce two concepts of "consistency" in the ordinal sense: a strong one and a weak one. Specifically, a subset X' is strongly consistent means that the ordering of their pairwise distances is the same under each of the input metric ρ_i from M. The weak consistency, on the other hand, relaxes this exact ordering condition, and intuitively allows us to take the plurality of ordering relation between two pairwise distances. We show in this paper that the maximum consensus problems over both the strong and the weak consistency notions are NP-complete, even when there are only 2 or 3 simple metrics, such as line metrics and ultrametrics. We also develop constant-factor approximation algorithms for the dual version, the minimum inconsistent subset problem of a metric system (M, P), - note that optimizing these two dual problems are equivalent.

cs.CC↗

arXiv2021

Graph Coarsening with Neural Networks

As large-scale graphs become increasingly more prevalent, it poses significant computational challenges to process, extract and analyze large graph data. Graph coarsening is one popular technique to reduce the size of a graph while maintaining essential properties. Despite rich graph coarsening literature, there is only limited exploration of data-driven methods in the field. In this work, we leverage the recent progress of deep learning on graphs for graph coarsening. We first propose a framework for measuring the quality of coarsening algorithm and show that depending on the goal, we need to carefully choose the Laplace operator on the coarse graph and associated projection/lift operators. Motivated by the observation that the current choice of edge weight for the coarse graph may be sub-optimal, we parametrize the weight assignment map with graph neural networks and train it to improve the coarsening quality in an unsupervised way. Through extensive experiments on both synthetic and real networks, we demonstrate that our method significantly improves common graph coarsening methods under various metrics, reduction ratios, graph sizes, and graph types. It generalizes to graphs of larger size ($25\times$ of training graphs), is adaptive to different losses (differentiable and non-differentiable), and scales to much larger graphs than previous work.

cs.LG↗

arXiv2020

A new algorithm for computing $μ$-bases of the univariate polynomial vector

In this paper, we characterized the relationship between Groebner bases and u-bases: any minimal Groebner basis of the syzygy module for n univariate polynomials with respect to the term-over-position monomial order is its u-basis. Moreover, based on the gcd computation, we construct a free basis of the syzygy module by the recursive way. According to this relationship and the constructed free basis, a new algorithm for computing u-bases of the syzygy module is presented. The theoretical complexity of the algorithm is O(n^3d^2) under a reasonable assumption, where d is the maximum degree of the input n polynomials. We have implemented this algorithm (MinGb) in Maple. Experimental data and performance comparison with the existing algorithms developed by Song and Goldman (2009) (SG algorithm) and Hong et al. (2017) (HHK algorithm) show that MinGb algorithm is more efficient than SG algorithm when n and d are sufficiently large, while MinGb algorithm and HHK algorithm both have their own advantages.

cs.SC↗

arXiv2020

On Minor Left Prime Factorization Problem for Multivariate Polynomial Matrices

A new necessary and sufficient condition for the existence of minor left prime factorizations of multivariate polynomial matrices without full row rank is presented. The key idea is to establish a relationship between a matrix and its full row rank submatrix. Based on the new result, we propose an algorithm for factorizing matrices and have implemented it on the computer algebra system Maple. Two examples are given to illustrate the effectiveness of the algorithm, and experimental data shows that the algorithm is efficient.

cs.SC↗

arXiv2020

On Factor Left Prime Factorization Problems for Multivariate Polynomial Matrices

This paper is concerned with factor left prime factorization problems for multivariate polynomial matrices without full row rank. We propose a necessary and sufficient condition for the existence of factor left prime factorizations of a class of multivariate polynomial matrices, and then design an algorithm to compute all factor left prime factorizations if they exist. We implement the algorithm on the computer algebra system Maple, and two examples are given to illustrate the effectiveness of the algorithm. The results presented in this paper are also true for the existence of factor right prime factorizations of multivariate polynomial matrices without full column rank.

cs.SC↗

arXiv2020

New Remarks on the Factorization and Equivalence Problems for a Class of Multivariate Polynomial Matrices

This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present some new criteria for the existence of matrix factorizations for a class of multivariate polynomial matrices, and obtain a necessary and sufficient condition for the equivalence of a square polynomial matrix and a diagonal matrix. Based on the constructive proof of the new criteria, we give a factorization algorithm and prove the uniqueness of the factorization. We implement the algorithm on Maple, and two illustrative examples are given to show the effectiveness of the algorithm.

cs.SC↗

arXiv2020

Detection and skeletonization of single neurons and tracer injections using topological methods

Neuroscientific data analysis has traditionally relied on linear algebra and stochastic process theory. However, the tree-like shapes of neurons cannot be described easily as points in a vector space (the subtraction of two neuronal shapes is not a meaningful operation), and methods from computational topology are better suited to their analysis. Here we introduce methods from Discrete Morse (DM) Theory to extract the tree-skeletons of individual neurons from volumetric brain image data, and to summarize collections of neurons labelled by tracer injections. Since individual neurons are topologically trees, it is sensible to summarize the collection of neurons using a consensus tree-shape that provides a richer information summary than the traditional regional 'connectivity matrix' approach. The conceptually elegant DM approach lacks hand-tuned parameters and captures global properties of the data as opposed to previous approaches which are inherently local. For individual skeletonization of sparsely labelled neurons we obtain substantial performance gains over state-of-the-art non-topological methods (over 10% improvements in precision and faster proofreading). The consensus-tree summary of tracer injections incorporates the regional connectivity matrix information, but in addition captures the collective collateral branching patterns of the set of neurons connected to the injection site, and provides a bridge between single-neuron morphology and tracer-injection data.

cs.CV↗

arXiv2019

Factorizations for a Class of Multivariate Polynomial Matrices

Following the works by Lin et al. (Circuits Syst. Signal Process. 20(6): 601-618, 2001) and Liu et al. (Circuits Syst. Signal Process. 30(3): 553-566, 2011), we investigate how to factorize a class of multivariate polynomial matrices. The main theorem in this paper shows that an $l\times m$ polynomial matrix admits a factorization with respect to a polynomial if the polynomial and all the $(l-1)\times (l-1)$ reduced minors of the matrix generate the unit ideal. This result is a further generalization of previous works, and based on this, we give an algorithm which can be used to factorize more polonomial matrices. In addition, an illustrate example is given to show that our main theorem is non-trivial and valuable.

cs.SC↗

arXiv2018

An Improved Cost Function for Hierarchical Cluster Trees

Hierarchical clustering has been a popular method in various data analysis applications. It partitions a data set into a hierarchical collection of clusters, and can provide a global view of (cluster) structure behind data across different granularity levels. A hierarchical clustering (HC) of a data set can be naturally represented by a tree, called a HC-tree, where leaves correspond to input data and subtrees rooted at internal nodes correspond to clusters. Many hierarchical clustering algorithms used in practice are developed in a procedure manner. Dasgupta proposed to study the hierarchical clustering problem from an optimization point of view, and introduced an intuitive cost function for similarity-based hierarchical clustering with nice properties as well as natural approximation algorithms. We observe that while Dasgupta's cost function is effective at differentiating a good HC-tree from a bad one for a fixed graph, the value of this cost function does not reflect how well an input similarity graph is consistent to a hierarchical structure. In this paper, we present a new cost function, which is developed based on Dasgupta's cost function, to address this issue. The optimal tree under the new cost function remains the same as the one under Dasgupta's cost function. However, the value of our cost function is more meaningful. The new way of formulating the cost function also leads to a polynomial time algorithm to compute the optimal cluster tree when the input graph has a perfect HC-structure, or an approximation algorithm when the input graph 'almost' has a perfect HC-structure. Finally, we provide further understanding of the new cost function by studying its behavior for random graphs sampled from an edge probability matrix.

cs.DS↗

arXiv2014

A Monomial-Oriented GVW for Computing Gröbner Bases

The GVW algorithm, presented by Gao et al., is a signature-based algorithm for computing Gröbner bases. In this paper, a variant of GVW is presented. This new algorithm is called a monomial-oriented GVW algorithm or mo-GVW algorithm for short. The mo-GVW algorithm presents a new frame of GVW and regards {\em labeled monomials} instead of {\em labeled polynomials} as basic elements of the algorithm. Being different from the original GVW algorithm, for each labeled monomial, the mo-GVW makes efforts to find the smallest signature that can generate this monomial. The mo-GVW algorithm also avoids generating J-pairs, and uses efficient methods of searching reducers and checking criteria. Thus, the mo-GVW algorithm has a better performance during practical implementations.

cs.SC↗