arXiv subjects

Yusu Wang

Publications and source records attributed to Yusu Wang.

At least 37 records · Page 2Linked to original sources

arXiv2024

Position: Topological Deep Learning is the New Frontier for Relational Learning

Topological deep learning (TDL) is a rapidly evolving field that uses topological features to understand and design deep learning models. This paper posits that TDL is the new frontier for relational learning. TDL may complement graph representation learning and geometric deep learning by incorporating topological concepts, and can thus provide a natural choice for various machine learning settings. To this end, this paper discusses open problems in TDL, ranging from practical benefits to theoretical foundations. For each problem, it outlines potential solutions and future research opportunities. At the same time, this paper serves as an invitation to the scientific community to actively participate in TDL research to unlock the potential of this emerging field.

cs.LG↗

arXiv2024

Principal Component Analysis in Space Forms

Principal Component Analysis (PCA) is a workhorse of modern data science. While PCA assumes the data conforms to Euclidean geometry, for specific data types, such as hierarchical and cyclic data structures, other spaces are more appropriate. We study PCA in space forms; that is, those with constant curvatures. At a point on a Riemannian manifold, we can define a Riemannian affine subspace based on a set of tangent vectors. Finding the optimal low-dimensional affine subspace for given points in a space form amounts to dimensionality reduction. Our Space Form PCA (SFPCA) seeks the affine subspace that best represents a set of manifold-valued points with the minimum projection cost. We propose proper cost functions that enjoy two properties: (1) their optimal affine subspace is the solution to an eigenequation, and (2) optimal affine subspaces of different dimensions form a nested set. These properties provide advances over existing methods, which are mostly iterative algorithms with slow convergence and weaker theoretical guarantees. We evaluate the proposed SFPCA on real and simulated data in spherical and hyperbolic spaces. We show that it outperforms alternative methods in estimating true subspaces (in simulated data) with respect to convergence speed or accuracy, often both.

stat.ML↗

arXiv2024

DE-HNN: An effective neural model for Circuit Netlist representation

The run-time for optimization tools used in chip design has grown with the complexity of designs to the point where it can take several days to go through one design cycle which has become a bottleneck. Designers want fast tools that can quickly give feedback on a design. Using the input and output data of the tools from past designs, one can attempt to build a machine learning model that predicts the outcome of a design in significantly shorter time than running the tool. The accuracy of such models is affected by the representation of the design data, which is usually a netlist that describes the elements of the digital circuit and how they are connected. Graph representations for the netlist together with graph neural networks have been investigated for such models. However, the characteristics of netlists pose several challenges for existing graph learning frameworks, due to the large number of nodes and the importance of long-range interactions between nodes. To address these challenges, we represent the netlist as a directed hypergraph and propose a Directional Equivariant Hypergraph Neural Network (DE-HNN) for the effective learning of (directed) hypergraphs. Theoretically, we show that our DE-HNN can universally approximate any node or hyperedge based function that satisfies certain permutation equivariant and invariant properties natural for directed hypergraphs. We compare the proposed DE-HNN with several State-of-the-art (SOTA) machine learning models for (hyper)graphs and netlists, and show that the DE-HNN significantly outperforms them in predicting the outcome of optimized place-and-route tools directly from the input netlists. Our source code and the netlists data used are publicly available at https://github.com/YusuLab/chips.git

cs.LG↗

arXiv2024

On the Theoretical Expressive Power and the Design Space of Higher-Order Graph Transformers

Graph transformers have recently received significant attention in graph learning, partly due to their ability to capture more global interaction via self-attention. Nevertheless, while higher-order graph neural networks have been reasonably well studied, the exploration of extending graph transformers to higher-order variants is just starting. Both theoretical understanding and empirical results are limited. In this paper, we provide a systematic study of the theoretical expressive power of order-$k$ graph transformers and sparse variants. We first show that, an order-$k$ graph transformer without additional structural information is less expressive than the $k$-Weisfeiler Lehman ($k$-WL) test despite its high computational cost. We then explore strategies to both sparsify and enhance the higher-order graph transformers, aiming to improve both their efficiency and expressiveness. Indeed, sparsification based on neighborhood information can enhance the expressive power, as it provides additional information about input graph structures. In particular, we show that a natural neighborhood-based sparse order-$k$ transformer model is not only computationally efficient, but also expressive -- as expressive as $k$-WL test. We further study several other sparse graph attention models that are computationally efficient and provide their expressiveness analysis. Finally, we provide experimental results to show the effectiveness of the different sparsification strategies.

cs.LG↗

arXiv2024

Distances for Markov Chains, and Their Differentiation

(Directed) graphs with node attributes are a common type of data in various applications and there is a vast literature on developing metrics and efficient algorithms for comparing them. Recently, in the graph learning and optimization communities, a range of new approaches have been developed for comparing graphs with node attributes, leveraging ideas such as the Optimal Transport (OT) and the Weisfeiler-Lehman (WL) graph isomorphism test. Two state-of-the-art representatives are the OTC distance proposed in (O'Connor et al., 2022) and the WL distance in (Chen et al., 2022). Interestingly, while these two distances are developed based on different ideas, we observe that they both view graphs as Markov chains, and are deeply connected. Indeed, in this paper, we propose a unified framework to generate distances for Markov chains (thus including (directed) graphs with node attributes), which we call the Optimal Transport Markov (OTM) distances, that encompass both the OTC and the WL distances. We further introduce a special one-parameter family of distances within our OTM framework, called the discounted WL distance. We show that the discounted WL distance has nice theoretical properties and can address several limitations of the existing OTC and WL distances. Furthermore, contrary to the OTC and the WL distances, our new discounted WL distance can be differentiated after a entropy-regularization similar to the Sinkhorn distance, making it suitable to use in learning frameworks, e.g., as the reconstruction loss in a graph generative model.

cs.LG↗

arXiv2024

Learning Ultrametric Trees for Optimal Transport Regression

Optimal transport provides a metric which quantifies the dissimilarity between probability measures. For measures supported in discrete metric spaces, finding the optimal transport distance has cubic time complexity in the size of the space. However, measures supported on trees admit a closed-form optimal transport that can be computed in linear time. In this paper, we aim to find an optimal tree structure for a given discrete metric space so that the tree-Wasserstein distance approximates the optimal transport distance in the original space. One of our key ideas is to cast the problem in ultrametric spaces. This helps us optimize over the space of ultrametric trees -- a mixed-discrete and continuous optimization problem -- via projected gradient decent over the space of ultrametric matrices. During optimization, we project the parameters to the ultrametric space via a hierarchical minimum spanning tree algorithm, equivalent to the closest projection to ultrametrics under the supremum norm. Experimental results on real datasets show that our approach outperforms previous approaches (e.g. Flowtree, Quadtree) in approximating optimal transport distances. Finally, experiments on synthetic data generated on ground truth trees show that our algorithm can accurately uncover the underlying trees.

cs.LG↗

arXiv2023

NN-Steiner: A Mixed Neural-algorithmic Approach for the Rectilinear Steiner Minimum Tree Problem

Recent years have witnessed rapid advances in the use of neural networks to solve combinatorial optimization problems. Nevertheless, designing the "right" neural model that can effectively handle a given optimization problem can be challenging, and often there is no theoretical understanding or justification of the resulting neural model. In this paper, we focus on the rectilinear Steiner minimum tree (RSMT) problem, which is of critical importance in IC layout design and as a result has attracted numerous heuristic approaches in the VLSI literature. Our contributions are two-fold. On the methodology front, we propose NN-Steiner, which is a novel mixed neural-algorithmic framework for computing RSMTs that leverages the celebrated PTAS algorithmic framework of Arora to solve this problem (and other geometric optimization problems). Our NN-Steiner replaces key algorithmic components within Arora's PTAS by suitable neural components. In particular, NN-Steiner only needs four neural network (NN) components that are called repeatedly within an algorithmic framework. Crucially, each of the four NN components is only of bounded size independent of input size, and thus easy to train. Furthermore, as the NN component is learning a generic algorithmic step, once learned, the resulting mixed neural-algorithmic framework generalizes to much larger instances not seen in training. Our NN-Steiner, to our best knowledge, is the first neural architecture of bounded size that has capacity to approximately solve RSMT (and variants). On the empirical front, we show how NN-Steiner can be implemented and demonstrate the effectiveness of our resulting approach, especially in terms of generalization, by comparing with state-of-the-art methods (both neural and non-neural based).

cs.AI↗

arXiv2023

Cycle Invariant Positional Encoding for Graph Representation Learning

Cycles are fundamental elements in graph-structured data and have demonstrated their effectiveness in enhancing graph learning models. To encode such information into a graph learning framework, prior works often extract a summary quantity, ranging from the number of cycles to the more sophisticated persistence diagram summaries. However, more detailed information, such as which edges are encoded in a cycle, has not yet been used in graph neural networks. In this paper, we make one step towards addressing this gap, and propose a structure encoding module, called CycleNet, that encodes cycle information via edge structure encoding in a permutation invariant manner. To efficiently encode the space of all cycles, we start with a cycle basis (i.e., a minimal set of cycles generating the cycle space) which we compute via the kernel of the 1-dimensional Hodge Laplacian of the input graph. To guarantee the encoding is invariant w.r.t. the choice of cycle basis, we encode the cycle information via the orthogonal projector of the cycle basis, which is inspired by BasisNet proposed by Lim et al. We also develop a more efficient variant which however requires that the input graph has a unique shortest cycle basis. To demonstrate the effectiveness of the proposed module, we provide some theoretical understandings of its expressive power. Moreover, we show via a range of experiments that networks enhanced by our CycleNet module perform better in various benchmarks compared to several existing SOTA models.

cs.LG↗

arXiv2023

Neural approximation of Wasserstein distance via a universal architecture for symmetric and factorwise group invariant functions

Learning distance functions between complex objects, such as the Wasserstein distance to compare point sets, is a common goal in machine learning applications. However, functions on such complex objects (e.g., point sets and graphs) are often required to be invariant to a wide variety of group actions e.g. permutation or rigid transformation. Therefore, continuous and symmetric product functions (such as distance functions) on such complex objects must also be invariant to the product of such group actions. We call these functions symmetric and factor-wise group invariant (or SFGI functions in short). In this paper, we first present a general neural network architecture for approximating SFGI functions. The main contribution of this paper combines this general neural network with a sketching idea to develop a specific and efficient neural network which can approximate the $p$-th Wasserstein distance between point sets. Very importantly, the required model complexity is independent of the sizes of input point sets. On the theoretical front, to the best of our knowledge, this is the first result showing that there exists a neural network with the capacity to approximate Wasserstein distance with bounded model complexity. Our work provides an interesting integration of sketching ideas for geometric problems with universal approximation of symmetric functions. On the empirical front, we present a range of results showing that our newly proposed neural network architecture performs comparatively or better than other models (including a SOTA Siamese Autoencoder based approach). In particular, our neural network generalizes significantly better and trains much faster than the SOTA Siamese AE. Finally, this line of investigation could be useful in exploring effective neural network design for solving a broad range of geometric optimization problems (e.g., $k$-means in a metric space).

cs.LG↗

arXiv2023

Universal Representation of Permutation-Invariant Functions on Vectors and Tensors

A main object of our study is multiset functions -- that is, permutation-invariant functions over inputs of varying sizes. Deep Sets, proposed by \cite{zaheer2017deep}, provides a \emph{universal representation} for continuous multiset functions on scalars via a sum-decomposable model. Restricting the domain of the functions to finite multisets of $D$-dimensional vectors, Deep Sets also provides a \emph{universal approximation} that requires a latent space dimension of $O(N^D)$ -- where $N$ is an upper bound on the size of input multisets. In this paper, we strengthen this result by proving that universal representation is guaranteed for continuous and discontinuous multiset functions though a latent space dimension of $O(N^D)$. We then introduce \emph{identifiable} multisets for which we can uniquely label their elements using an identifier function, namely, finite-precision vectors are identifiable. Using our analysis on identifiable multisets, we prove that a sum-decomposable model for general continuous multiset functions only requires a latent dimension of $2DN$. We further show that both encoder and decoder functions of the model are continuous -- our main contribution to the existing work which lack such a guarantee. Also this provides a significant improvement over the aforementioned $O(N^D)$ bound which was derived for universal representation of continuous and discontinuous multiset functions. We then extend our results and provide special sum-decomposition structures to universally represent permutation-invariant tensor functions on identifiable tensors. These families of sum-decomposition models enables us to design deep network architectures and deploy them on a variety of learning tasks on sequences, images, and graphs.

cs.LG↗

arXiv2023

The Weisfeiler-Lehman Distance: Reinterpretation and Connection with GNNs

In this paper, we present a novel interpretation of the so-called Weisfeiler-Lehman (WL) distance, introduced by Chen et al. (2022), using concepts from stochastic processes. The WL distance aims at comparing graphs with node features, has the same discriminative power as the classic Weisfeiler-Lehman graph isomorphism test and has deep connections to the Gromov-Wasserstein distance. This new interpretation connects the WL distance to the literature on distances for stochastic processes, which also makes the interpretation of the distance more accessible and intuitive. We further explore the connections between the WL distance and certain Message Passing Neural Networks, and discuss the implications of the WL distance for understanding the Lipschitz property and the universal approximation results for these networks.

cs.LG↗

arXiv2023

Minimum Monotone Tree Decomposition of Density Functions Defined on Graphs

Monotone trees - trees with a function defined on their vertices that decreases the further away from a root node one travels, are a natural model for a process that weakens the further one gets from its source. Given an aggregation of monotone trees, one may wish to reconstruct the individual monotone components. A natural representation of such an aggregation would be a graph. While many methods have been developed for extracting hidden graph structure from datasets, which makes obtaining such an aggregation possible, decomposing such graphs into the original monotone trees is algorithmically challenging. Recently, a polynomial time algorithm has been developed to extract a minimum cardinality collection of monotone trees (M-Tree Set) from a given density tree - but no such algorithm exists for density graphs that may contain cycles. In this work, we prove that extracting such minimum M-Tree Sets of density graphs is NP-Complete. We additionally prove three additional variations of the problem - such as the minimum M-Tree Set such that the intersection between any two monotone trees is either empty or contractible (SM-Tree Set) - are also NP-Complete. We conclude by providing some approximation algorithms, highlighted by a 3-approximation algorithm for computing the minimum SM-Tree Set for density cactus graphs.

cs.DS↗

arXiv2023

Random Walks, Conductance, and Resistance for the Connection Graph Laplacian

We investigate the concept of effective resistance in connection graphs, expanding its traditional application from undirected graphs. We propose a robust definition of effective resistance in connection graphs by focusing on the duality of Dirichlet-type and Poisson-type problems on connection graphs. Additionally, we delve into random walks, taking into account both node transitions and vector rotations. This approach introduces novel concepts of effective conductance and resistance matrices for connection graphs, capturing mean rotation matrices corresponding to random walk transitions. Thereby, it provides new theoretical insights for network analysis and optimization.

cs.DM↗

arXiv2023

On the Connection Between MPNN and Graph Transformer

Graph Transformer (GT) recently has emerged as a new paradigm of graph learning algorithms, outperforming the previously popular Message Passing Neural Network (MPNN) on multiple benchmarks. Previous work (Kim et al., 2022) shows that with proper position embedding, GT can approximate MPNN arbitrarily well, implying that GT is at least as powerful as MPNN. In this paper, we study the inverse connection and show that MPNN with virtual node (VN), a commonly used heuristic with little theoretical understanding, is powerful enough to arbitrarily approximate the self-attention layer of GT. In particular, we first show that if we consider one type of linear transformer, the so-called Performer/Linear Transformer (Choromanski et al., 2020; Katharopoulos et al., 2020), then MPNN + VN with only O(1) depth and O(1) width can approximate a self-attention layer in Performer/Linear Transformer. Next, via a connection between MPNN + VN and DeepSets, we prove the MPNN + VN with O(n^d) width and O(1) depth can approximate the self-attention layer arbitrarily well, where d is the input feature dimension. Lastly, under some assumptions, we provide an explicit construction of MPNN + VN with O(1) width and O(n) depth approximating the self-attention layer in GT arbitrarily well. On the empirical side, we demonstrate that 1) MPNN + VN is a surprisingly strong baseline, outperforming GT on the recently proposed Long Range Graph Benchmark (LRGB) dataset, 2) our MPNN + VN improves over early implementation on a wide range of OGB datasets and 3) MPNN + VN outperforms Linear Transformer and MPNN on the climate modeling task.

cs.LG↗

arXiv2023

Understanding Oversquashing in GNNs through the Lens of Effective Resistance

Message passing graph neural networks (GNNs) are a popular learning architectures for graph-structured data. However, one problem GNNs experience is oversquashing, where a GNN has difficulty sending information between distant nodes. Understanding and mitigating oversquashing has recently received significant attention from the research community. In this paper, we continue this line of work by analyzing oversquashing through the lens of the effective resistance between nodes in the input graph. Effective resistance intuitively captures the ``strength'' of connection between two nodes by paths in the graph, and has a rich literature spanning many areas of graph theory. We propose to use total effective resistance as a bound of the total amount of oversquashing in a graph and provide theoretical justification for its use. We further develop an algorithm to identify edges to be added to an input graph to minimize the total effective resistance, thereby alleviating oversquashing. We provide empirical evidence of the effectiveness of our total effective resistance based rewiring strategies for improving the performance of GNNs.

cs.LG↗

arXiv2023

Implicit Graphon Neural Representation

Graphons are general and powerful models for generating graphs of varying size. In this paper, we propose to directly model graphons using neural networks, obtaining Implicit Graphon Neural Representation (IGNR). Existing work in modeling and reconstructing graphons often approximates a target graphon by a fixed resolution piece-wise constant representation. Our IGNR has the benefit that it can represent graphons up to arbitrary resolutions, and enables natural and efficient generation of arbitrary sized graphs with desired structure once the model is learned. Furthermore, we allow the input graph data to be unaligned and have different sizes by leveraging the Gromov-Wasserstein distance. We first demonstrate the effectiveness of our model by showing its superior performance on a graphon learning task. We then propose an extension of IGNR that can be incorporated into an auto-encoder framework, and demonstrate its good performance under a more general setting of graphon learning. We also show that our model is suitable for graph representation learning and graph generation.

cs.LG↗

arXiv2023

A generalization of the persistent Laplacian to simplicial maps

The graph Laplacian is a fundamental object in the analysis of and optimization on graphs. This operator can be extended to a simplicial complex $K$ and therefore offers a way to perform ``signal processing" on $p$-(co)chains of $K$. Recently, the concept of persistent Laplacian was proposed and studied for a pair of simplicial complexes $K\hookrightarrow L$ connected by an inclusion relation, further broadening the use of Laplace-based operators. In this paper, we expand the scope of the persistent Laplacian by generalizing it to a pair of simplicial complexes connected by a simplicial map $f: K \to L$. Such simplicial map setting arises frequently, e.g., when relating a coarsened simplicial representation with an original representation, or the case when the two simplicial complexes are spanned by different point sets i.e. cases in which it does not hold that $K\subset L$. However, the simplicial map setting is more challenging than the inclusion setting since the underlying algebraic structure is more complicated. We present a natural generalization of the persistent Laplacian to the simplicial setting. To shed insight on the structure behind it, as well as to develop an algorithm to compute it, we exploit the relationship between the persistent Laplacian and the Schur complement of a matrix. A critical step is to view the Schur complement as a functorial way of restricting a self-adjoint PSD operator to a given subspace. As a consequence, we prove that persistent Betti numbers of a simplicial map can be recovered by persistent Laplacians. We then propose an algorithm for finding the matrix representations of persistent Laplacians which in turn yields a new algorithm for computing persistent Betti numbers of a simplicial map. Finally, we study the persistent Laplacian on simplicial towers under simplicial maps and establish monotonicity results for their eigenvalues.

math.CO↗

arXiv2022

Convergence of Invariant Graph Networks

Although theoretical properties such as expressive power and over-smoothing of graph neural networks (GNN) have been extensively studied recently, its convergence property is a relatively new direction. In this paper, we investigate the convergence of one powerful GNN, Invariant Graph Network (IGN) over graphs sampled from graphons. We first prove the stability of linear layers for general $k$-IGN (of order $k$) based on a novel interpretation of linear equivariant layers. Building upon this result, we prove the convergence of $k$-IGN under the model of \citet{ruiz2020graphon}, where we access the edge weight but the convergence error is measured for graphon inputs. Under the more natural (and more challenging) setting of \citet{keriven2020convergence} where one can only access 0-1 adjacency matrix sampled according to edge probability, we first show a negative result that the convergence of any IGN is not possible. We then obtain the convergence of a subset of IGNs, denoted as IGN-small, after the edge probability estimation. We show that IGN-small still contains function class rich enough that can approximate spectral GNNs arbitrarily well. Lastly, we perform experiments on various graphon models to verify our statements.

cs.LG↗