arXiv subjects

Zhiqiang Xu

Publications and source records attributed to Zhiqiang Xu.

At least 37 records · Page 2Linked to original sources

arXiv2026

Improving Enzyme Prediction with Chemical Reaction Equations by Hypergraph-Enhanced Knowledge Graph Embeddings

Predicting enzyme-substrate interactions has long been a fundamental problem in biochemistry and metabolic engineering. While existing methods could leverage databases of expert-curated enzyme-substrate pairs for models to learn from known pair interactions, the databases are often sparse, i.e., there are only limited and incomplete examples of such pairs, and also labor-intensive to maintain. This lack of sufficient training data significantly hinders the ability of traditional enzyme prediction models to generalize to unseen interactions. In this work, we try to exploit chemical reaction equations from domain-specific databases, given their easier accessibility and denser, more abundant data. However, interactions of multiple compounds, e.g., educts and products, with the same enzymes create complex relational data patterns that traditional models cannot easily capture. To tackle that, we represent chemical reaction equations as triples of (educt, enzyme, product) within a knowledge graph, such that we can take advantage of knowledge graph embedding (KGE) to infer missing enzyme-substrate pairs for graph completion. Particularly, in order to capture intricate relationships among compounds, we propose our knowledge-enhanced hypergraph model for enzyme prediction, i.e., Hyper-Enz, which integrates a hypergraph transformer with a KGE model to learn representations of the hyper-edges that involve multiple educts and products. Also, a multi-expert paradigm is introduced to guide the learning of enzyme-substrate interactions with both the proposed model and chemical reaction equations. Experimental results show a significant improvement, with up to a 88% relative improvement in average enzyme retrieval accuracy and 30% improvement in pair-level prediction compared to traditional models, demonstrating the effectiveness of our approach.

cs.AI↗

arXiv2025

Sharpness-aware Federated Graph Learning

One of many impediments to applying graph neural networks (GNNs) to large-scale real-world graph data is the challenge of centralized training, which requires aggregating data from different organizations, raising privacy concerns. Federated graph learning (FGL) addresses this by enabling collaborative GNN model training without sharing private data. However, a core challenge in FGL systems is the variation in local training data distributions among clients, known as the data heterogeneity problem. Most existing solutions suffer from two problems: (1) The typical optimizer based on empirical risk minimization tends to cause local models to fall into sharp valleys and weakens their generalization to out-of-distribution graph data. (2) The prevalent dimensional collapse in the learned representations of local graph data has an adverse impact on the classification capacity of the GNN model. To this end, we formulate a novel optimization objective that is aware of the sharpness (i.e., the curvature of the loss surface) of local GNN models. By minimizing the loss function and its sharpness simultaneously, we seek out model parameters in a flat region with uniformly low loss values, thus improving the generalization over heterogeneous data. By introducing a regularizer based on the correlation matrix of local representations, we relax the correlations of representations generated by individual local graph samples, so as to alleviate the dimensional collapse of the learned model. The proposed \textbf{S}harpness-aware f\textbf{E}derated gr\textbf{A}ph \textbf{L}earning (SEAL) algorithm can enhance the classification accuracy and generalization ability of local GNN models in federated graph learning. Experimental studies on several graph classification benchmarks show that SEAL consistently outperforms SOTA FGL baselines and provides gains for more participants.

cs.LG↗

arXiv2025

Generalized Interlacing Families: New Error Bounds for CUR Matrix Decompositions

This paper introduces the concept of generalized interlacing families of polynomials, which extends the classical interlacing polynomial method to handle polynomials of varying degrees. We establish a fundamental property for these families, proving the existence of a polynomial with a desired degree whose smallest root is greater than or equal to the smallest root of the expected polynomial. Applying this framework to the generalized CUR matrix approximation problem, we derive a theoretical upper bound on the spectral norm of a residual matrix, expressed in terms of the largest root of the expected polynomial. We then explore two important special cases: the classical CUR matrix decompositions and the row subset selection problem. For classical CUR matrix decompositions, we derive an explicit upper bound for the largest root of the expected polynomial. This yields a tighter spectral norm error bound for the residual matrix compared to many existing results. Furthermore, we present a deterministic polynomial-time algorithm for solving the classical CUR problem under certain matrix conditions. For the row subset selection problem, we establish the first known spectral norm error bound. This paper extends the applicability of interlacing families and deepens the theoretical foundations of CUR matrix decompositions and related approximation problems.

math.RA↗

arXiv2025

Utility Boundary of Dataset Distillation: Scaling and Configuration-Coverage Laws

Dataset distillation (DD) aims to construct compact synthetic datasets that allow models to achieve comparable performance to full-data training while substantially reducing storage and computation. Despite rapid empirical progress, its theoretical foundations remain limited: existing methods (gradient, distribution, trajectory matching) are built on heterogeneous surrogate objectives and optimization assumptions, which makes it difficult to analyze their common principles or provide general guarantees. Moreover, it is still unclear under what conditions distilled data can retain the effectiveness of full datasets when the training configuration, such as optimizer, architecture, or augmentation, changes. To answer these questions, we propose a unified theoretical framework, termed configuration--dynamics--error analysis, which reformulates major DD approaches under a common generalization-error perspective and provides two main results: (i) a scaling law that provides a single-configuration upper bound, characterizing how the error decreases as the distilled sample size increases and explaining the commonly observed performance saturation effect; and (ii) a coverage law showing that the required distilled sample size scales linearly with configuration diversity, with provably matching upper and lower bounds. In addition, our unified analysis reveals that various matching methods are interchangeable surrogates, reducing the same generalization error, clarifying why they can all achieve dataset distillation and providing guidance on how surrogate choices affect sample efficiency and robustness. Experiments across diverse methods and configurations empirically confirm the derived laws, advancing a theoretical foundation for DD and enabling theory-driven design of compact, configuration-robust dataset distillation.

cs.LG↗

arXiv2025

Local-Curvature-Aware Knowledge Graph Embedding: An Extended Ricci Flow Approach

Knowledge graph embedding (KGE) relies on the geometry of the embedding space to encode semantic and structural relations. Existing methods place all entities on one homogeneous manifold, Euclidean, spherical, hyperbolic, or their product/multi-curvature variants, to model linear, symmetric, or hierarchical patterns. Yet a predefined, homogeneous manifold cannot accommodate the sharply varying curvature that real-world graphs exhibit across local regions. Since this geometry is imposed a priori, any mismatch with the knowledge graph's local curvatures will distort distances between entities and hurt the expressiveness of the resulting KGE. To rectify this, we propose RicciKGE to have the KGE loss gradient coupled with local curvatures in an extended Ricci flow such that entity embeddings co-evolve dynamically with the underlying manifold geometry towards mutual adaptation. Theoretically, when the coupling coefficient is bounded and properly selected, we rigorously prove that i) all the edge-wise curvatures decay exponentially, meaning that the manifold is driven toward the Euclidean flatness; and ii) the KGE distances strictly converge to a global optimum, which indicates that geometric flattening and embedding optimization are promoting each other. Experimental improvements on link prediction and node classification benchmarks demonstrate RicciKGE's effectiveness in adapting to heterogeneous knowledge graph structures.

cs.LG↗

arXiv2025

GeoDM: Geometry-aware Distribution Matching for Dataset Distillation

Dataset distillation aims to synthesize a compact subset of the original data, enabling models trained on it to achieve performance comparable to those trained on the original large dataset. Existing distribution-matching methods are confined to Euclidean spaces, making them only capture linear structures and overlook the intrinsic geometry of real data, e.g., curvature. However, high-dimensional data often lie on low-dimensional manifolds, suggesting that dataset distillation should have the distilled data manifold aligned with the original data manifold. In this work, we propose a geometry-aware distribution-matching framework, called \textbf{GeoDM}, which operates in the Cartesian product of Euclidean, hyperbolic, and spherical manifolds, with flat, hierarchical, and cyclical structures all captured by a unified representation. To adapt to the underlying data geometry, we introduce learnable curvature and weight parameters for three kinds of geometries. At the same time, we design an optimal transport loss to enhance the distribution fidelity. Our theoretical analysis shows that the geometry-aware distribution matching in a product space yields a smaller generalization error bound than the Euclidean counterparts. Extensive experiments conducted on standard benchmarks demonstrate that our algorithm outperforms state-of-the-art data distillation methods and remains effective across various distribution-matching strategies for the single geometries.

cs.CV↗

arXiv2025

Instance optimality in phase retrieval

Compressed sensing has demonstrated that a general signal $\boldsymbol{x} \in \mathbb{F}^n$ ($\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$) can be estimated from few linear measurements with an error {proportional to} the best $k$-term approximation error, a property known as instance optimality. In this paper, we investigate instance optimality in the context of phaseless measurements using the $\ell_p$-minimization decoder, where $p \in (0, 1]$, for both real and complex cases. More specifically, we prove that $(2,1)$ and $(1,1)$-instance optimality of order $k$ can be achieved with $m =O(k \log(n/k))$ phaseless measurements, paralleling results from linear measurements. These results imply that one can stably recover approximately $k$-sparse signals from $m = O(k \log(n/k))$ phaseless measurements. Our approach leverages the phaseless bi-Lipschitz condition. Additionally, we present a non-uniform version of $(2,2)$-instance optimality result in probability applicable to any fixed vector $\boldsymbol{x} \in \mathbb{F}^n$. These findings reveal striking parallels between compressive phase retrieval and classical compressed sensing, enhancing our understanding of both phase retrieval and instance optimality.

math.FA↗

arXiv2025

Concise Reasoning in the Lens of Lagrangian Optimization

Concise reasoning in large language models seeks to generate only essential intermediate steps needed to arrive at a final answer, thereby alleviating issues of overthinking. Most proposed approaches hinge on carefully hand-crafted heuristics, struggling to balance concision with performance, often failing to adapt across domains and model scales. In this work, we address these challenges by introducing a principled and pragmatic strategy, performance-aware length updating (PALU). As a principled algorithm, PALU formulates concise reasoning as a constrained optimization problem, minimizing response length subject to a performance constraint, and then applies Lagrangian optimization to convert it into a tractable unconstrained problem. As a pragmatic solution, PALU streamlines complicated update rules through three approximations: (i) estimating performance with off-policy rollouts, (ii) truncating the Lagrange multiplier to two extremes, and (iii) replacing gradient-based updates with quantile-driven length adjustments. PALU reduces output length by 65% while improving accuracy by 15% when applied to DeepSeek-Distill-Qwen-1.5B, averaged over five benchmarks, outperforming a range of alternative methods. Furthermore, PALU is demonstrated to adapt across both domain (logic, STEM and math) and model scale (1.5B, 7B, 14B) entrenching the algorithm as a practical and effective concise reasoning approach.

cs.AI↗

arXiv2025

Human Texts Are Outliers: Detecting LLM-generated Texts via Out-of-distribution Detection

The rapid advancement of large language models (LLMs) such as ChatGPT, DeepSeek, and Claude has significantly increased the presence of AI-generated text in digital communication. This trend has heightened the need for reliable detection methods to distinguish between human-authored and machine-generated content. Existing approaches both zero-shot methods and supervised classifiers largely conceptualize this task as a binary classification problem, often leading to poor generalization across domains and models. In this paper, we argue that such a binary formulation fundamentally mischaracterizes the detection task by assuming a coherent representation of human-written texts. In reality, human texts do not constitute a unified distribution, and their diversity cannot be effectively captured through limited sampling. This causes previous classifiers to memorize observed OOD characteristics rather than learn the essence of `non-ID' behavior, limiting generalization to unseen human-authored inputs. Based on this observation, we propose reframing the detection task as an out-of-distribution (OOD) detection problem, treating human-written texts as distributional outliers while machine-generated texts are in-distribution (ID) samples. To this end, we develop a detection framework using one-class learning method including DeepSVDD and HRN, and score-based learning techniques such as energy-based method, enabling robust and generalizable performance. Extensive experiments across multiple datasets validate the effectiveness of our OOD-based approach. Specifically, the OOD-based method achieves 98.3% AUROC and AUPR with only 8.9% FPR95 on DeepFake dataset. Moreover, we test our detection framework on multilingual, attacked, and unseen-model and -domain text settings, demonstrating the robustness and generalizability of our framework. Code, pretrained weights, and demo will be released.

cs.CL↗

arXiv2025

On the Performance of Amplitude-Based Models for Low-Rank Matrix Recovery

In this paper, we focus on low-rank phase retrieval, which aims to reconstruct a matrix $\mathbf{X}_0\in \mathbb{R}^{n\times m}$ with ${\mathrm{ rank}}(\mathbf{X}_0)\le r$ from noise-corrupted amplitude measurements $\mathbf{y}=|\mathcal{A}(\mathbf{X}_0)|+\boldsymbolη$, where $\mathcal{A}:\mathbb{R}^{n\times m}\rightarrow \mathbb{R}^{p}$ is a linear map and $\boldsymbolη\in \mathbb{R}^p$ is the noise vector. We first examine the rank-constrained nonlinear least-squares model $\hat{\mathbf{X}}\in \mathop{\mathrm{argmin}}\limits_{\substack{\mathbf{X}\in \mathbb{R}^{n\times m},\mathrm{rank}(\mathbf{X})\le r}}\||\mathcal{A}(\mathbf{X})|-\mathbf{y}\|_2^2$ to estimate $\mathbf{X}_0$, and demonstrate that the reconstruction error satisfies $\min\{\|\hat{\mathbf{X}}-\mathbf{X}_0\|_F, \|\hat{\mathbf{X}}+\mathbf{X}_0\|_F\}\lesssim \frac{\|\boldsymbolη\|_2}{\sqrt{p}}$ with high probability, provided $\mathcal{A}$ is a Gaussian measurement ensemble and $p\gtrsim (m+n)r$. We also prove that the error bound $\frac{\|\boldsymbolη\|_2}{\sqrt{p}}$ is tight up to a constant. Furthermore, we relax the rank constraint to a nuclear-norm constraint. Hence, we propose the Lasso model for low-rank phase retrieval, i.e., the constrained nuclear-norm model and the unconstrained version. We also establish comparable theoretical guarantees for these models. To achieve this, we introduce a strong restricted isometry property (SRIP) for the linear map $\mathcal{A}$, analogous to the strong RIP in phase retrieval. This work provides a unified treatment that extends existing results in both phase retrieval and low-rank matrix recovery from rank-one measurements.

cs.IT↗

arXiv2025

New Lower Bounds for the Minimum Singular Value in Matrix Selection

The objective of the matrix selection problem is to select a submatrix $A_{S}\in \mathbb{R}^{n\times k}$ from $A\in \mathbb{R}^{n\times m}$ such that its minimum singular value is maximized. In this paper, we employ the interlacing polynomial method to investigate this problem. This approach allows us to identify a submatrix $A_{S_0}\in \mathbb{R}^{n\times k}$ and establish a lower bound for its minimum singular value. Specifically, unlike common interlacing polynomial approaches that estimate the smallest root of the expected characteristic polynomial via barrier functions, we leverage the direct relationship between roots and coefficients. This leads to a tighter lower bound when $k$ is close to $n$. For the case where $AA^{\top}=\mathbb{I}_n$ and $k=n$, our result improves the well-known result by Hong-Pan, which involves extracting a basis from a tight frame and establishing a lower bound for the minimum singular value of the basis matrix.

math.FA↗

arXiv2025

Can Optical Denoising Clean Sonar Images? A Benchmark and Fusion Approach

Object detection in sonar images is crucial for underwater robotics applications including autonomous navigation and resource exploration. However, complex noise patterns inherent in sonar imagery, particularly speckle, reverberation, and non-Gaussian noise, significantly degrade detection accuracy. While denoising techniques have achieved remarkable success in optical imaging, their applicability to sonar data remains underexplored. This study presents the first systematic evaluation of nine state-of-the-art deep denoising models with distinct architectures, including Neighbor2Neighbor with varying noise parameters, Blind2Unblind with different noise configurations, and DSPNet, for sonar image preprocessing. We establish a rigorous benchmark using five publicly available sonar datasets and assess their impact on four representative detection algorithms: YOLOX, Faster R-CNN, SSD300, and SSDMobileNetV2. Our evaluation addresses three unresolved questions: first, how effectively optical denoising architectures transfer to sonar data; second, which model families perform best against sonar noise; and third, whether denoising truly improves detection accuracy in practical pipelines. Extensive experiments demonstrate that while denoising generally improves detection performance, effectiveness varies across methods due to their inherent biases toward specific noise types. To leverage complementary denoising effects, we propose a mutually-supervised multi-source denoising fusion framework where outputs from different denoisers mutually supervise each other at the pixel level, creating a synergistic framework that produces cleaner images.

cs.CV↗

arXiv2025

Golden Noise for Diffusion Models: A Learning Framework

Text-to-image diffusion model is a popular paradigm that synthesizes personalized images by providing a text prompt and a random Gaussian noise. While people observe that some noises are ``golden noises'' that can achieve better text-image alignment and higher human preference than others, we still lack a machine learning framework to obtain those golden noises. To learn golden noises for diffusion sampling, we mainly make three contributions in this paper. First, we identify a new concept termed the \textit{noise prompt}, which aims at turning a random Gaussian noise into a golden noise by adding a small desirable perturbation derived from the text prompt. Following the concept, we first formulate the \textit{noise prompt learning} framework that systematically learns ``prompted'' golden noise associated with a text prompt for diffusion models. Second, we design a noise prompt data collection pipeline and collect a large-scale \textit{noise prompt dataset}~(NPD) that contains 100k pairs of random noises and golden noises with the associated text prompts. With the prepared NPD as the training dataset, we trained a small \textit{noise prompt network}~(NPNet) that can directly learn to transform a random noise into a golden noise. The learned golden noise perturbation can be considered as a kind of prompt for noise, as it is rich in semantic information and tailored to the given text prompt. Third, our extensive experiments demonstrate the impressive effectiveness and generalization of NPNet on improving the quality of synthesized images across various diffusion models, including SDXL, DreamShaper-xl-v2-turbo, and Hunyuan-DiT. Moreover, NPNet is a small and efficient controller that acts as a plug-and-play module with very limited additional inference and computational costs, as it just provides a golden noise instead of a random noise without accessing the original pipeline.

cs.LG↗

arXiv2025

Signal Recovery on Algebraic Varieties Using Linear Samples

The recovery of an unknown signal from its linear measurements is a fundamental problem spanning numerous scientific and engineering disciplines. Commonly, prior knowledge suggests that the underlying signal resides within a known algebraic variety. This context naturally leads to a question: what is the minimum number of measurements required to uniquely recover any signal belonging to such an algebraic variety? In this survey paper, we introduce a method that leverages tools from algebraic geometry to address this question. We then demonstrate the utility of this approach by applying it to two problems: phase retrieval and low-rank matrix recovery. We also highlight several open problems, which could serve as a basis for future investigations in this field.

cs.IT↗

arXiv2025

Principled Data Selection for Alignment: The Hidden Risks of Difficult Examples

The alignment of large language models (LLMs) often assumes that using more clean data yields better outcomes, overlooking the match between model capacity and example difficulty. Challenging this, we propose a new principle: Preference data vary in difficulty, and overly difficult examples hinder alignment, by exceeding the model's capacity. Through systematic experimentation, we validate this principle with three key findings: (1) preference examples vary in difficulty, as evidenced by consistent learning orders across alignment runs; (2) overly difficult examples significantly degrade performance across four LLMs and two datasets; and (3) the capacity of a model dictates its threshold for handling difficult examples, underscoring a critical relationship between data selection and model capacity. Building on this principle, we introduce Selective DPO, which filters out overly difficult examples. This simple adjustment improves alignment performance by 9-16% in win rates on the AlpacaEval 2 benchmark compared to the DPO baseline, suppressing a series of DPO variants with different algorithmic adjustments. Together, these results illuminate the importance of aligning data difficulty with model capacity, offering a transformative perspective for improving alignment strategies in LLMs. Code is available at https://github.com/glorgao/SelectiveDPO.

cs.CL↗

arXiv2025

Rethinking Soft Actor-Critic in High-Dimensional Action Spaces: The Cost of Ignoring Distribution Shift

Soft Actor-Critic algorithm is widely recognized for its robust performance across a range of deep reinforcement learning tasks, where it leverages the tanh transformation to constrain actions within bounded limits. However, this transformation induces a distribution shift, distorting the original Gaussian action distribution and potentially leading the policy to select suboptimal actions, particularly in high-dimensional action spaces. In this paper, we conduct a comprehensive theoretical and empirical analysis of this distribution shift, deriving the precise probability density function (PDF) for actions following the tanh transformation to clarify the misalignment introduced between the transformed distribution's mode and the intended action output. We substantiate these theoretical insights through extensive experiments on high-dimensional tasks within the HumanoidBench benchmark. Our findings indicate that accounting for this distribution shift substantially enhances SAC's performance, resulting in notable improvements in cumulative rewards, sample efficiency, and reliability across tasks. These results underscore a critical consideration for SAC and similar algorithms: addressing transformation-induced distribution shifts is essential to optimizing policy effectiveness in high-dimensional deep reinforcement learning environments, thereby expanding the robustness and applicability of SAC in complex control tasks.

cs.LG↗

arXiv2025

Interlacing Polynomial Method for Matrix Approximation via Generalized Column and Row Selection

This paper delves into the spectral norm aspect of the Generalized Column and Row Subset Selection (GCRSS) problem. Given a target matrix $\mathbf{A}\in \mathbb{R}^{n\times d}$, the objective of GCRSS is to select a column submatrix $\mathbf{B}_{:,S}\in\mathbb{R}^{n\times k}$ from the source matrix $\mathbf{B}\in\mathbb{R}^{n\times d_B}$ and a row submatrix $\mathbf{C}_{R,:}\in\mathbb{R}^{r\times d}$ from the source matrix $\mathbf{C}\in\mathbb{R}^{n_C\times d}$, such that the residual matrix $(\mathbf{I}_n-\mathbf{B}_{:,S}\mathbf{B}_{:,S}^{\dagger})\mathbf{A}(\mathbf{I}_d-\mathbf{C}_{R,:}^{\dagger} \mathbf{C}_{R,:})$ has a small spectral norm. By employing the method of interlacing polynomials, we show that the smallest possible spectral norm of a residual matrix can be bounded by the largest root of a related expected characteristic polynomial. A deterministic polynomial time algorithm is provided for the spectral norm case of the GCRSS problem. We next focus on two specific GCRSS scenarios: the Generalized Column Subset Selection (GCSS) problem ($r=0$), and the submatrix selection problem ($\mathbf{B}=\mathbf{C}=\mathbf{I}_d$). In the GCSS scenario, we connect the expected characteristic polynomials to the convolution of multi-affine polynomials, leading to the derivation of the first provable reconstruction bound on the spectral norm of a residual matrix. In the submatrix selection scenario, we show that for any sufficiently small $\varepsilon>0$ and any square matrix $\mathbf{A}\in\mathbb{R}^{d\times d}$, there exist two subsets $S\subset [d]$ and $R\subset [d]$ of sizes $O(d\cdot \varepsilon^2)$ such that $\Vert\mathbf{A}_{S,R}\Vert_2\leq \varepsilon\cdot \Vert\mathbf{A}\Vert_2$.

math.FA↗

arXiv2025

MagicDistillation: Weak-to-Strong Video Distillation for Large-Scale Few-Step Synthesis

Recently, open-source video diffusion models (VDMs), such as WanX, Magic141 and HunyuanVideo, have been scaled to over 10 billion parameters. These large-scale VDMs have demonstrated significant improvements over smaller-scale VDMs across multiple dimensions, including enhanced visual quality and more natural motion dynamics. However, these models face two major limitations: (1) High inference overhead: Large-scale VDMs require approximately 10 minutes to synthesize a 28-step video on a single H100 GPU. (2) Limited in portrait video synthesis: Models like WanX-I2V and HunyuanVideo-I2V often produce unnatural facial expressions and movements in portrait videos. To address these challenges, we propose MagicDistillation, a novel framework designed to reduce inference overhead while ensuring the generalization of VDMs for portrait video synthesis. Specifically, we primarily use sufficiently high-quality talking video to fine-tune Magic141, which is dedicated to portrait video synthesis. We then employ LoRA to effectively and efficiently fine-tune the fake DiT within the step distillation framework known as distribution matching distillation (DMD). Following this, we apply weak-to-strong (W2S) distribution matching and minimize the discrepancy between the fake data distribution and the ground truth distribution, thereby improving the visual fidelity and motion dynamics of the synthesized videos. Experimental results on portrait video synthesis demonstrate the effectiveness of MagicDistillation, as our method surpasses Euler, LCM, and DMD baselines in both FID/FVD metrics and VBench. Moreover, MagicDistillation, requiring only 4 steps, also outperforms WanX-I2V (14B) and HunyuanVideo-I2V (13B) on visualization and VBench. Our project page is https://magicdistillation.github.io/MagicDistillation/.

cs.CV↗