arXiv subjects

Chunfeng Cui

Publications and source records attributed to Chunfeng Cui.

At least 19 recordsLinked to original sources

arXiv2026

A Weak Condition for Limited Augmented Zarankiewicz Numbers

This paper introduces the weak augmented Zarankiewicz number $z_{wA}(m,n)$ and the weak limited augmented Zarankiewicz number $z_{wL}(m,n)$, which are combinatorial extensions of the classical Zarankiewicz number obtained by relaxing the original admissibility conditions for augmented bipartite graphs. We show that the resulting weak framework still guarantees irreducibility of the associated doubly simple biquadratic forms, with SOS rank equal to the total number of edges. This yields the inequality chain \[ \mathrm{BSR}(m,n) \geq z_{wA}(m,n) \geq z_{wL}(m,n) \geq z_L(m,n) \geq z(m,n). \] We provide three complementary constructions demonstrating the power of the weak framework. First, a $5\times 3$ construction using degenerate 2-edges yields $z_{wL}(5,3)\ge 10>9=z_L(5,3)$, giving $\mathrm{BSR}(5,3)\ge 10$. Second, a $15\times 6$ construction on the incidence graph of $K_6$ with 14 nondegenerate 2-edges gives $z_{wL}(15,6)\ge 44>43$, improving the previously known bound. Third, a critical $6\times 3$ construction with complementary 2-cycles gives $z_{wL}(6,3)\ge 12>11=z_L(6,3)$, yielding $\mathrm{BSR}(6,3)\ge 12$ and demonstrating that complementary 2-cycles are safe.

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arXiv2026

MoSSP: A Momentum-Based Single-Loop Stochastic Penalty Method for Nonconvex Constrained DC-Regularized Optimization

In this paper, we study a structured class of nonconvex constrained stochastic problems with difference-of-convex (DC) regularization, where the feasible set is possibly nonconvex and the concave part of the DC regularizer is allowed to be nonsmooth. The fundamental challenge lies in maintaining feasibility for nonconvex constraints while achieving favorable oracle complexity. Although single-loop algorithms efficiently solve unconstrained DC optimization problems, their potential for constrained optimization with DC structure remains largely unexplored. To address this gap, we develop MoSSP, a Momentum-based Single-loop Stochastic Penalty method for such problems with provable complexity guarantees. The key idea is to apply a single stochastic proximal-gradient step to the Moreau envelope of the penalty plus the convex DC part, with the concave part's proximal mapping computed in parallel. We derive two algorithm variants: a Polyak-momentum version with $O(\varepsilon^{-4})$ oracle complexity for finding stochastic $\varepsilon$-KKT points, and an improved $O(\varepsilon^{-3})$ version incorporating recursive momentum. Experimental results demonstrate the effectiveness of the proposed algorithms.

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arXiv2026

Three-Edges and the SOS Rank of Biquadratic Forms

We extend the augmented bipartite graph framework for biquadratic sum-of-squares (SOS) ranks by introducing $3$-edges -- triples of cells representing squares of three-term bilinear forms. We define suitable generalized cycle-free conditions that are purely combinatorial yet sufficient to guarantee that the SOS rank equals the total number of edges, carefully distinguishing occupation by $1$/$2$-edges from occupation by $3$-edges. The main theorem states that for any generalized cycle-free augmented bipartite graph $G$ satisfying the simplicity condition (S), the associated triply simple biquadratic form $P_G$ satisfies $\operatorname{sos}(P_G) = |E_1| + |E_2| + |E_3|$. The proof extends the orthogonality method with a novel trick: when a $2$-edge and a $3$-edge interact, the $3$-edge condition must be invoked rather than the $2$-edge condition. We give three applications. A $5 \times 3$ construction with two $2$-edges, inadmissible under the original definition, is admissible under our new definition, yielding $z_{3L}(5,3) \ge 10$ and improving $z_L(5,3)=9$. A $10 \times 5$ graph using a column-fully-degenerate $3$-edge gives $z_{3L}(10,5) \ge 27$, separating it from $z_L(10,5)=26$. A $15 \times 6$ graph using a half-row-degenerate $3$-edge improves the lower bound for $\operatorname{BSR}(15,6)$ from $43$ to $44$. These are the first explicit applications of $3$-edges to obtain improved lower bounds for $\operatorname{BSR}(m,n)$, and the $5 \times 3$ example demonstrates the power of the refined conditions.

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arXiv2026

A General Lower Bound for the Limited Augmented Zarankiewicz Number based upon Complete Graphs

The limited augmented Zarankiewicz number $z_L(m,n)$ satisfies $\operatorname{BSR}(m,n)\ge z_L(m,n)\ge z(m,n)$, where $\operatorname{BSR}(m,n)$ is the maximum SOS rank of $m\times n$ biquadratic forms and $z(m,n)$ is the classical Zarankiewicz number. Our main result is a general lower bound for $z_L(m,n)$ based on the incidence graph of the complete graph $K_{4t}$. For every integer $t\ge 1$, let $m = \binom{4t}{2}$ and $n = 4t$. Then $$ \operatorname{BSR}(m,n) \;\ge\; z_L(m,n) \;\ge\; 2\binom{4t}{2} + 4t^2 - 2t. $$ Since $z = 2\binom{4t}{2} = \Theta(t^2)$, the gap satisfies $z_L - z \ge 4t^2 - 2t = \Theta(t^2) = \Theta(m)$, i.e., it grows linearly in $m$. Moreover, $$ \frac{z_L - z}{z} \;\ge\; \frac{4t^2}{16t^2 - 4t} \;\longrightarrow\; \frac{1}{4} \quad \text{as } t\to\infty, $$ so the gap is asymptotically at least $25\%$ of $z$ -- a non-negligible constant fraction. For $t=1$ we obtain $z_L(6,4)\ge 14$, and we prove that this bound is tight, i.e., $z_L(6,4)=14$. For $t=2$ and $t=3$ we obtain $z_L(28,8)\ge 68$ and $z_L(66,12)\ge 162$, respectively, improving previously known bounds. We also determine the exact values of $z_L(m,n)$ for $5\times3$ and $5\times4$: $z_L(5,3)=9$ and $z_L(5,4)=12$. These results serve as base cases for a \emph{lifting method} that constructs admissible limited augmented graphs on $(m+1)\times(n+1)$ from optimal ones on $m\times n$. Applying this method, we obtain new lower bounds: $z_L(6,3)\ge 10$ and $z_L(6,5)\ge 17$. For $5\times 5$ we establish a new lower bound $z_L(5,5)\ge 15$, improving the previously known bound. By a direct construction, we have $z_L(6,6)\ge 20$.

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arXiv2026

Biquadratic SOS Rank and Augmented Zarankiewicz Number

This paper introduces the concepts of the augmented Zarankiewicz number $z_A(m,n)$ and the limited augmented Zarankiewicz number $z_L(m,n)$, which are natural combinatorial extensions of the classical Zarankiewicz number. These numbers arise from augmented bipartite graphs that may contain both standard edges (1-edges) and pairs of edges representing squares of binomials (2-edges). The main theoretical result establishes the inequality chain $\operatorname{BSR}(m, n) \geq z_A(m, n) \geq z_L(m, n) \geq z(m, n)$, linking the maximum biquadratic sum-of-squares (SOS) rank to these extremal graph parameters. We determine the exact values of $z_L(m, n)$ for the cases $(m,2)$, $(3,3)$, $(4, 3)$ and $(4,4)$, and provide new lower bounds for the cases $(5,3)$, $(5,4)$, and $(5,5)$. These results yield improved lower bounds for the maximum SOS rank of biquadratic forms, demonstrating that $z_L(m,n)$ can exceed the classical Zarankiewicz number, thereby offering a refined combinatorial perspective on the SOS rank problem.

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arXiv2026

Narrowing the Gap: SOS Ranks of $4 \times 3$ Biquadratic Forms and a Lower Bound of $8$

We investigate the maximum sum-of-squares (SOS) rank of biquadratic forms in the critical case of $4 \times 3$ variables, where the general bounds are currently $7 \leq \mathrm{BSR}(4,3) \leq 11$. By analyzing two important structured subclasses, we obtain exact determinations and improved upper bounds that significantly narrow this gap. For simple biquadratic forms those containing only distinct terms of the type $x_i^2 y_j^2$ we prove that the maximum achievable SOS rank is exactly 7, a value attained by a form corresponding to a $C_4$-free bipartite graph with the maximum number of edges. This settles the question for simple forms. For $y$-deficient biquadratic forms a class introduced here that permits cross terms among two of the three $y$-variables while the third appears only in pure square terms we prove an upper bound of $9$ by combining Calder\"{o}n's theorem on $m\times 2$ forms with the known value $\mathrm{BSR}(4,2) = 5$. Our main result is a constructive proof that $\mathrm{BSR}(4,3) \geq 8$. We present an explicit non-simple, non-deficient $4\times 3$ biquadratic form and prove it requires exactly eight squares, thereby improving the general lower bound. This shows that any form achieving a rank higher than $8$ must possess a more complex algebraic structure, and it reduces the search space for determining the true value of $\mathrm{BSR}(4,3)$. Connections to Zarankiewicz numbers, extremal graph theory, and classical results on sums of squares are highlighted throughout.

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arXiv2026

Sum of Squares Rank of Biquadratic Forms and The Zarankiewicz Number

Denote the maximum sos rank of $m \times n$ sum of squares (SOS) biquadratic forms by $BSR(m, n)$. In this paper, we show that $BSR(m, n) \ge z(m, n)$ and conjecture that $BSR(m, n) = z(m, n)$, where $z(m, n)$ is the Zarankiewicz number. Our result coincides with the existing results for $m = 2$, $n = 2$, and $m = n = 3$, and is superior to other previously known lower bounds. Our result also connects graph theory and SOS polynomial theory.

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arXiv2026

Parallelizable Riemannian Alternating Direction Method of Multipliers for Non-convex Pose Graph Optimization

Pose graph optimization (PGO) is fundamental to robot perception and navigation systems, serving as the mathematical backbone for solving simultaneous localization and mapping (SLAM). Existing solvers suffer from polynomial growth in computational complexity with graph size, hindering real-time deployment in large-scale scenarios. In this paper, by duplicating variables and introducing equality constraints, we reformulate the problem and propose a Parallelizable Riemannian Alternating Direction Method of Multipliers (PRADMM) to solve it efficiently. Compared with the state-of-the-art methods that usually exhibit polynomial time complexity growth with graph size, PRADMM enables efficient parallel computation across vertices regardless of graph size. Crucially, all subproblems admit closed-form solutions, ensuring PRADMM maintains exceptionally stable performance. Furthermore, by carefully exploiting the structures of the coefficient matrices in the constraints, we establish the global convergence of PRADMM under mild conditions, enabling larger relaxation step sizes within the interval $(0,2)$. Extensive empirical validation on two synthetic datasets and multiple real-world 3D SLAM benchmarks confirms the superior computational performance of PRADMM.

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arXiv2026

Sum of Squares Decompositions and Rank Bounds for Biquadratic Forms

We study SOS properties of biquadratic forms. For the class of partially symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness and prove that every PSD partially symmetric biquadratic form is a sum of squares of bilinear forms. This extends the known result for fully symmetric biquadratic forms. We describe an efficient computational procedure for constructing SOS decompositions, exploiting the Kronecker-product structure of the associated matrix representation. We introduce simple biquadratic forms. For $m \ge 2$, we present a $m \times 2$ PSD biquadratic form and show that it can be expressed as the sum of $m+1$ squares, but cannot be expressed as the sum of $m$ squares. This provides a lower bound for sos rank of $m \times 2$ biquadratic forms, and shows that previously proved results that a $2 \times 2$ PSD biquadratic form can be expressed as the sum of three squares, and a $3 \times 2$ PSD biquadratic form can be expressed as the sum of four squares, are tight. We also present an $3 \times 3$ SOS biquadratic form, which can be expressed as the sum of six squares, but not the sum of five squares.We present a $2 \times 2$ PSD biquadratic form, and show that it can be expressed as the sum of three squares, but cannot be expressed as the sum of two squares. Furthermore, we present a $3 \times 2$ PSD biquadratic form, and show that it can be expressed as the sum of four squares, but cannot be expressed as the sum of three squares. These show that previously proved results that a $2 \times 2$ PSD biquadratic form can be expressed as the sum of three squares, and a $3 \times 2$ PSD biquadratic form can be expressed as the sum of four squares, are tight. Moreover, we establish a universal upper bound SOS-rank$(P) \le mn-1$ for any SOS biquadratic form, which improves the trivial bound $mn$ and is tight in small dimensions.

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arXiv2025

Sum of Squares Decompositions for Structured Biquadratic Forms

This paper studies sum-of-squares (SOS) representations for structured biquadratic forms. We prove that diagonally dominated symmetric biquadratic tensors are always SOS. For the special case of symmetric biquadratic forms, we establish necessary and sufficient conditions for positive semi-definiteness of monic symmetric biquadratic forms, characterize the geometry of the corresponding PSD cone as a convex polyhedron, and prove that every such PSD form is SOS for any dimensions $m$ and $n$. We also formulate conjectures regarding SOS representations for symmetric M-biquadratic tensors and symmetric $\mathrm{B}_{0}$-biquadratic tensors, discussing their likelihood and potential proof strategies. Our results advance the understanding of when positive semi-definiteness implies sum-of-squares decompositions for structured biquadratic forms.

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arXiv2025

The Power Method for Non-Hermitian Dual Quaternion Matrices

This paper proposes a power method for computing the dominant eigenvalues of a non-Hermitian dual quaternion matrix (DQM). Although the algorithmic framework parallels the Hermitian case, the theoretical analysis is substantially more complex since a non-Hermitian dual matrix may possess no eigenvalues or infinitely many eigenvalues. Besides, its eigenvalues are not necessarily dual numbers, leading to non-commutative behavior that further complicates the analysis. We first present a sufficient condition that ensures the existence of an eigenvalue whose standard part corresponds to the largest magnitude eigenvalue of the standard part matrix. Under a stronger condition, we then establish that the sequence generated by the power method converges linearly to the strict dominant eigenvalue and its associated eigenvectors. We also verify that this condition is necessary. The key to our analysis is a new Jordan-like decomposition, which addresses a gap arising from the lack of a conventional Jordan decomposition for non-Hermitian dual matrices. Our framework readily extends to non-Hermitian dual complex and dual number matrices. We also develop an adjoint method that reformulates the eigenvalue problem into an equivalent form for dual complex matrices. Numerical experiments on non-Hermitian DQMs are presented to demonstrate the efficiency of the power method.

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arXiv2025

iFCTN: an intra-block Fully-Connected Tensor Network Decomposition for Tensor Completion

The fully-connected tensor network (FCTN) decomposition has recently exhibited strong modeling capabilities by connecting every pair of tensor factors, thereby capturing rich cross-mode correlations. However, this advantage comes with an inherent limitation: updating the factors typically requires reconstructing auxiliary sub-networks, which entails extensive and cumbersome (un)folding. In this study, we propose the intra-block FCTN (iFCTN) decomposition, a novel (un)folding-free variant of FCTN decomposition designed to enhance computational efficiency. We parameterize each FCTN factor through Khatri-Rao products, which significantly reduces the complexity of reconstructing intermediate sub-networks and yields subproblems with well-structured coefficient matrices. Furthermore, we deploy the proposed iFCTN decomposition on the representative task of tensor completion and design an efficient proximal alternating minimization algorithm. Theoretically, we establish its global convergence to a critical point. Extensive experiments demonstrate that iFCTN outperforms state-of-the-art methods with a lower computational overhead.

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arXiv2025

Structured Symmetric Tensors

In this paper, we study structured symmetric tensors. We introduce several new classes of structured symmetric tensors: completely decomposable (CD) tensors, strictly sum of squares (SSOS) tensors and SOS$^*$ tensors. CD tensors have applications in data analysis and signal processing. Complete Hankel tensors are CD tensors. SSOS tensors are defined as SOS tensors with a positive definite Gram matrix, ensuring structural stability under perturbations. The SOS$^*$ cone is defined as the dual cone of the SOS tensor cone, with characterizations via moment matrices and polynomial nonnegativity. We study the relations among completely positive (CP) cones, CD cones, sum of squares (SOS) cones, positive semidefinite (PSD) cones and copositive (COP) cones. We identify the interiors of PSD, SOS, CP, COP and CD cones for even-order tensors. These characterizations are crucial for interior-point methods and stability analysis in polynomial and tensor optimization. We generalize the classical Schur product theorem to CD and CP tensors, including the case of strongly completely decomposable (SCD) and strongly completely positive (SCP) tensors. We identify equivalence between strictly CD (SCD) and positive definite (PD) for CD tensors. Furthermore, we give an example of a PSD but not SOS Hankel tensor. This answers an open question raised in the literature.

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arXiv2025

Projecting onto the unit dual quaternion set

Dual quaternions have gained significant attention due to their wide applications in areas such as multi-agent formation control, 3D motion modeling, and robotics. A fundamental aspect in dual quaternion research involves the projection onto the unit dual quaternion set. In this paper, we systematically study such projections under the $2^R$-norm, which is commonly used in practical applications. We identify several distinct cases based on the relationship between the standard and dual parts in vector form, and demonstrate the effectiveness of the proposed algorithm through numerical experiments.

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arXiv2025

DualHash: A Stochastic Primal-Dual Algorithm with Theoretical Guarantee for Deep Hashing

Deep hashing converts high-dimensional feature vectors into compact binary codes, enabling efficient large-scale retrieval. A fundamental challenge in deep hashing stems from the discrete nature of quantization in generating the codes. W-type regularizations, such as $||z|-1|$, have been proven effective as they encourage variables toward binary values. However, existing methods often directly optimize these regularizations without convergence guarantees. While proximal gradient methods offer a promising solution, the coupling between W-type regularizers and neural network outputs results in composite forms that generally lack closed-form proximal solutions. In this paper, we present a stochastic primal-dual hashing algorithm, referred to as DualHash, that provides rigorous complexity bounds. Using Fenchel duality, we partially transform the nonconvex W-type regularization optimization into the dual space, which results in a proximal operator that admits closed-form solutions. We derive two algorithm instances: a momentum-accelerated version with $\mathcal{O}(\varepsilon^{-4})$ complexity and an improved $\mathcal{O}(\varepsilon^{-3})$ version using variance reduction. Experiments on three image retrieval databases demonstrate the superior performance of DualHash.

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arXiv2025

Completely Positive Biquadratic Tensors

In this paper, we systemically introduce completely positive biquadratic (CPB) tensors and copositive biquadratic tensors. We show that all weakly CPB tensors are sum of squares tensors, the CPB tensor cone and the copositive biquadratic tensor cone are dual cone to each other. We also show that the outer product of two completely positive matrices is a CPB tensor, and the outer product of two copositive matrices is a copositive biquadratic tensor. We then study two easily checkable subclasses of CPB tensors, namely positive biquadratic Cauchy tensors and biquadratic Pascal tensors. We show that a biquadratic Pascal tensor is both strongly CPB and positive definite.

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arXiv2025

A Dual Quaternion Control Law for Formation Control of Multiple 3-D Rigid Bodies

This paper studies the integrated position and attitude control problem for multi-agent systems of 3D rigid bodies. While the state-of-the-art method in [Olfati-Saber and Murray, 2004] established the theoretical foundation for rigid-body formation control, it requires all agents to asymptotically converge to identical positions and attitudes, limiting its applicability in scenarios where distinct desired relative configurations must be maintained. In this paper, we develop a novel dual-quaternion-based framework that generalizes this paradigm. By introducing a unit dual quaternion directed graph (UDQDG) representation, we derive a new control law through the corresponding Laplacian matrix, enabling simultaneous position and attitude coordination while naturally accommodating directed interaction topologies. Leveraging the recent advances in UDQDG spectral theory, we prove global asymptotic convergence to desired relative configurations modulo a right-multiplicative constant and establish an R-linear convergence rate determined by the second smallest eigenvalue of the UDQDG Laplacian. A projected iteration method is proposed to compute the iterative states. Finally, the proposed solution is verified by several numerical experiments.

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arXiv2025

The SOS Rank of Biquadratic Forms

In 1973, Calder\'{o}n proved that an $m \times 2$ positive semidefinite (psd) biquadratic form can always be expressed as the sum of ${3m(m+1) \over 2}$ squares of quadratic forms. Very recently, by applying Hilbert's theorem on ternary quartics, we proved that a $2 \times 2$ psd biquadratic form can always be expressed as the sum of three squares of bilinear forms. This improved Calder\'{o}n's result for $m=2$, and left the sos (sum-of-squares) rank problem of $m \times 2$ biquadratic forms for $m \ge 3$ to further exploration. In this paper, we show that an $3 \times 2$ psd biquadratic form can always be expressed as four squares of bilinear forms. We make a conjecture that an $m \times 2$ psd biquadratic form can always be expressed as $m+1$ squares of bilinear forms.

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