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For some positive integer $m$, a real polynomial $P(x)=\sum\limits_{k=0}^ma_kx^k$ with $a_k\geqslant 0$ is called log-concave (resp. ultra log-concave) if $a_k^2\geqslant a_{k-1}a_{k+1}$ (resp. $a_k^2\geqslant \left(1+\frac{1}{k}\right)\left(1+\frac{1}{m-k}\right)\cdot$ $a_{k-1}a_{k+1}$) for all $1\leqslant k\leqslant m-1$. If $P(x)$ has only real roots, then it is called real-rooted. It is well-known that the conditions of log-concavity, ultra log-concavity and real-rootedness are ever-stronger. For a graph $G$, a dependent set is a set of vertices which is not independent, i.e., the set of vertices whose induced subgraph contains at least one edge. The dependence polynomial of $G$ is defined as $D(G, x):=\sum\limits_{k\geqslant 0}d_k(G)x^k$, where $d_k(G)$ is the number of dependent sets of size $k$ in $G$. Horrocks proved that $D(G, x)$ is log-concave for every graph $G$ [J. Combin. Theory, Ser. B, 84 (2002) 180--185]. In the present paper, we prove that, for a graph $G$, $D(G, x)$ is ultra log-concave if $G$ is $(K_2\cup 2K_1)$-free or contains an independent set of size $|V(G)|-2$, and give the characterization of graphs whose dependence polynomials are real-rooted. Finally, we focus more attention to the problems of log-concavity about independence systems and pose several conjectures closely related the famous Mason's Conjecture., Comment: 17 pages, 5 figures

Gradient-variation online learning aims to achieve regret guarantees that scale with the variations in the gradients of online functions, which has been shown to be crucial for attaining fast convergence in games and robustness in stochastic optimization, hence receiving increased attention. Existing results often require the smoothness condition by imposing a fixed bound on the gradient Lipschitzness, but this may not hold in practice. Recent efforts in neural network optimization suggest a generalized smoothness condition, allowing smoothness to correlate with gradient norms. In this paper, we systematically study gradient-variation online learning under generalized smoothness. To this end, we extend the classic optimistic mirror descent algorithm to derive gradient-variation bounds by conducting stability analysis over the optimization trajectory and exploiting smoothness locally. Furthermore, we explore universal online learning, designing a single algorithm enjoying optimal gradient-variation regrets for convex and strongly convex functions simultaneously without knowing curvature information. The algorithm adopts a two-layer structure with a meta-algorithm running over a group of base-learners. To ensure favorable guarantees, we have designed a new meta-algorithm that is Lipschitz-adaptive to handle potentially unbounded gradients and meanwhile ensures second-order regret to cooperate with base-learners. Finally, we provide implications of our findings and obtain new results in fast-rate games and stochastic extended adversarial optimization.

Image captioning evaluation metrics can be divided into two categories, reference-based metrics and reference-free metrics. However, reference-based approaches may struggle to evaluate descriptive captions with abundant visual details produced by advanced multimodal large language models, due to their heavy reliance on limited human-annotated references. In contrast, previous reference-free metrics have been proven effective via CLIP cross-modality similarity. Nonetheless, CLIP-based metrics, constrained by their solution of global image-text compatibility, often have a deficiency in detecting local textual hallucinations and are insensitive to small visual objects. Besides, their single-scale designs are unable to provide an interpretable evaluation process such as pinpointing the position of caption mistakes and identifying visual regions that have not been described. To move forward, we propose a novel reference-free metric for image captioning evaluation, dubbed Hierarchical Image Captioning Evaluation Score (HICE-S). By detecting local visual regions and textual phrases, HICE-S builds an interpretable hierarchical scoring mechanism, breaking through the barriers of the single-scale structure of existing reference-free metrics. Comprehensive experiments indicate that our proposed metric achieves the SOTA performance on several benchmarks, outperforming existing reference-free metrics like CLIP-S and PAC-S, and reference-based metrics like METEOR and CIDEr. Moreover, several case studies reveal that the assessment process of HICE-S on detailed captions closely resembles interpretable human judgments.Our code is available at https://github.com/joeyz0z/HICE., Comment: Accepted by ACM MM2024

Graph Neural Networks (GNNs) have shown impressive performance in graph representation learning, but they face challenges in capturing long-range dependencies due to their limited expressive power. To address this, Graph Transformers (GTs) were introduced, utilizing self-attention mechanism to effectively model pairwise node relationships. Despite their advantages, GTs suffer from quadratic complexity w.r.t. the number of nodes in the graph, hindering their applicability to large graphs. In this work, we present Graph-Enhanced Contextual Operator (GECO), a scalable and effective alternative to GTs that leverages neighborhood propagation and global convolutions to effectively capture local and global dependencies in quasilinear time. Our study on synthetic datasets reveals that GECO reaches 169x speedup on a graph with 2M nodes w.r.t. optimized attention. Further evaluations on diverse range of benchmarks showcase that GECO scales to large graphs where traditional GTs often face memory and time limitations. Notably, GECO consistently achieves comparable or superior quality compared to baselines, improving the SOTA up to 4.5%, and offering a scalable and effective solution for large-scale graph learning., Comment: Under submission

In this study, we constructed daily high-frequency sentiment data and used the VAR method to attempt to predict the next day's implied volatility surface. We utilized 630,000 text data entries from the East Money Stock Forum from 2014 to 2023 and employed deep learning methods such as BERT and LSTM to build daily market sentiment indicators. By applying FFT and EMD methods for sentiment decomposition, we found that high-frequency sentiment had a stronger correlation with at-the-money (ATM) options' implied volatility, while low-frequency sentiment was more strongly correlated with deep out-of-the-money (DOTM) options' implied volatility. Further analysis revealed that the shape of the implied volatility surface contains richer market sentiment information beyond just market panic. We demonstrated that incorporating this sentiment information can improve the accuracy of implied volatility surface predictions., Comment: 21 pages, 8 figures

Graph transformer has been proven as an effective graph learning method for its adoption of attention mechanism that is capable of capturing expressive representations from complex topological and feature information of graphs. Graph transformer conventionally performs dense attention (or global attention) for every pair of nodes to learn node representation vectors, resulting in quadratic computational costs that are unaffordable for large-scale graph data. Therefore, mini-batch training for graph transformers is a promising direction, but limited samples in each mini-batch can not support effective dense attention to encode informative representations. Facing this bottleneck, (1) we start by assigning each node a token list that is sampled by personalized PageRank (PPR) and then apply standard multi-head self-attention only on this list to compute its node representations. This PPR tokenization method decouples model training from complex graph topological information and makes heavy feature engineering offline and independent, such that mini-batch training of graph transformers is possible by loading each node's token list in batches. We further prove this PPR tokenization is viable as a graph convolution network with a fixed polynomial filter and jumping knowledge. However, only using personalized PageRank may limit information carried by a token list, which could not support different graph inductive biases for model training. To this end, (2) we rewire graphs by introducing multiple types of virtual connections through structure- and content-based super nodes that enable PPR tokenization to encode local and global contexts, long-range interaction, and heterophilous information into each node's token list, and then formalize our Virtual Connection Ranking based Graph Transformer (VCR-Graphormer).

Zero-shot image captioning (IC) without well-paired image-text data can be divided into two categories, training-free and text-only-training. Generally, these two types of methods realize zero-shot IC by integrating pretrained vision-language models like CLIP for image-text similarity evaluation and a pre-trained language model (LM) for caption generation. The main difference between them is whether using a textual corpus to train the LM. Though achieving attractive performance w.r.t. some metrics, existing methods often exhibit some common drawbacks. Training-free methods tend to produce hallucinations, while text-only-training often lose generalization capability. To move forward, in this paper, we propose a novel Memory-Augmented zero-shot image Captioning framework (MeaCap). Specifically, equipped with a textual memory, we introduce a retrieve-then-filter module to get key concepts that are highly related to the image. By deploying our proposed memory-augmented visual-related fusion score in a keywords-to-sentence LM, MeaCap can generate concept-centered captions that keep high consistency with the image with fewer hallucinations and more world-knowledge. The framework of MeaCap achieves the state-of-the-art performance on a series of zero-shot IC settings. Our code is available at https://github.com/joeyz0z/MeaCap., Comment: Accepted by CVPR2024

This paper presents a learnable solver tailored to iteratively solve sparse linear systems from discretized partial differential equations (PDEs). Unlike traditional approaches relying on specialized expertise, our solver streamlines the algorithm design process for a class of PDEs through training, which requires only training data of coefficient distributions. The proposed method is anchored by three core principles: (1) a multilevel hierarchy to promote rapid convergence, (2) adherence to linearity concerning the right-hand-side of equations, and (3) weights sharing across different levels to facilitate adaptability to various problem sizes. Built on these foundational principles and considering the similar computation pattern of the convolutional neural network (CNN) as multigrid components, we introduce a network adept at solving linear systems from PDEs with heterogeneous coefficients, discretized on structured grids. Notably, our proposed solver possesses the ability to generalize over right-hand-side terms, PDE coefficients, and grid sizes, thereby ensuring its training is purely offline. To evaluate its effectiveness, we train the solver on convection-diffusion equations featuring heterogeneous diffusion coefficients. The solver exhibits swift convergence to high accuracy over a range of grid sizes, extending from $31 \times 31$ to $4095 \times 4095$. Remarkably, our method outperforms the classical Geometric Multigrid (GMG) solver, demonstrating a speedup of approximately 3 to 8 times. Furthermore, our numerical investigation into the solver's capacity to generalize to untrained coefficient distributions reveals promising outcomes., Comment: 24 pages, 11 figures

Non-stationary online learning has drawn much attention in recent years. In particular, dynamic regret and adaptive regret are proposed as two principled performance measures for online convex optimization in non-stationary environments. To optimize them, a two-layer online ensemble is usually deployed due to the inherent uncertainty of the non-stationarity, in which a group of base-learners are maintained and a meta-algorithm is employed to track the best one on the fly. However, the two-layer structure raises the concern about the computational complexity -- those methods typically maintain $\mathcal{O}(\log T)$ base-learners simultaneously for a $T$-round online game and thus perform multiple projections onto the feasible domain per round, which becomes the computational bottleneck when the domain is complicated. In this paper, we present efficient methods for optimizing dynamic regret and adaptive regret, which reduce the number of projections per round from $\mathcal{O}(\log T)$ to $1$. Moreover, our obtained algorithms require only one gradient query and one function evaluation at each round. Our technique hinges on the reduction mechanism developed in parameter-free online learning and requires non-trivial twists on non-stationary online methods. Empirical studies verify our theoretical findings., Comment: preliminary conference version appeared at NeurIPS 2022; this extended version improves the paper presentation, further investigates the interval dynamic regret, and adds two applications (online non-stochastic control and online PCA)

Partial cubes are the graphs which can be embedded into hypercubes. The {\em cube polynomial} of a graph $G$ is a counting polynomial of induced hypercubes of $G$, which is defined as $C(G,x):=\sum_{i\geqslant 0}\alpha_i(G)x^i$, where $\alpha_i(G)$ is the number of induced $i$-cubes (hypercubes of dimension $i$) of $G$. The {\em clique polynomial} of $G$ is defined as $Cl(G,x):=\sum_{i\geqslant 0}a_i(G)x^i$, where $a_i(G)$ ($i\geqslant 1$) is the number of $i$-cliques in $G$ and $a_0(G)=1$. Equivalently, $Cl(G, x)$ is exactly the independence polynomial of the complement $\overline{G}$ of $G$. The {\em crossing graph} $G^{\#}$ of a partial cube $G$ is the graph whose vertices are corresponding to the $\Theta$-classes of $G$, and two $\Theta$-classes are adjacent in $G^{\#}$ if and only if they cross in $G$. In the present paper, we prove that for a partial cube $G$, $C(G,x)\leqslant Cl(G^{\#}, x+1)$ and the equality holds if and only if $G$ is a median graph. Since every graph can be represented as the crossing graph of a median graph [SIAM J. Discrete Math., 15 (2002) 235--251], the above necessary-and-sufficient result shows that the study on the cube polynomials of median graphs can be transformed to the one on the clique polynomials of general graphs (equivalently, on the independence polynomials of their complements). In addition, we disprove the conjecture that the cube polynomials of median graphs are unimodal., Comment: 13 pages,2 figures

Partial cubes are graphs that can be isometrically embedded into hypercubes. Convex cycles play an important role in the study of partial cubes. In this paper, we prove that a regular partial cube is a hypercube (resp., a doubled Odd graph, an even cycle of length $2n$ where $n\geqslant 4$) if and only if all its convex cycles are 4-cycles (resp., 6-cycles, $2n$-cycles). In particular, the partial cubes whose all convex cycles are 4-cycles are equivalent to almost-median graphs, so we obtain that regular almost-median graphs are exactly hypercubes, which generate the result by Mulder [J. Graph Theory, 4 (1980) 107--110] -- regular median graphs are hypercubes., Comment: 8 pages, 0 figures

Qubit memory that is entangled with photonic qubit is the building block for long distance quantum repeaters. Cavity enhanced and long lived spin wave photon entanglement has been demonstrated by applying dual laser beams onto optical-lattice atoms. However, owing to cross readouts by two beams, retrieval efficiency of spin wave qubit is decreased by one quarter compared to that of single mode spin wave at all storage times. Here, by coupling cold atoms to two modes of a polarization interferometer based cavity, we achieve perfect qubit retrieval in cavity enhanced and long lived atom photon entanglement. A write laser beam is applied onto cold atoms, we then create a magnetic field insensitive spin wave qubit that is entangled with the photonic qubit encoded onto two arms of the interferometer. The spin wave qubit is retrieved by a read beam, which avoids the cross readouts. Our experiment demonstrates 540{\mu}s storage time at 50% intrinsic qubit retrieval efficiency, which is 13.5 times longer than the best reported result.