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1. Rapid and Precise Topological Comparison with Merge Tree Neural Networks

   https://doi.org/10.1109/TVCG.2024.3456395  

   Qin, Yu; Fasy, Brittany Terese; Wenk, Carola; Summa, Brian (September 2024, IEEE Transactions on Visualization and Computer Graphics)

   Merge trees are a valuable tool in the scientific visualization of scalar fields; however, current methods for merge tree comparisons are computationally expensive, primarily due to the exhaustive matching between tree nodes. To address this challenge, we introduce the Merge Tree Neural Network (MTNN), a learned neural network model designed for merge tree comparison. The MTNN enables rapid and high-quality similarity computation. We first demonstrate how to train graph neural networks, which emerged as effective encoders for graphs, in order to to produce embeddings of merge trees in vector spaces for efficient similarity comparison. Next, we formulate the novel MTNN model that further improves the similarity comparisons by integrating the tree and node embeddings with a new topological attention mechanism. We demonstrate the effectiveness of our model on real-world data in different domains and examine our model's generalizability across various datasets. Our experimental analysis demonstrates our approach's superiority in accuracy and efficiency. In particular, we speed up the prior state-of-the-art by more than 100x on the benchmark datasets while maintaining an error rate below 0.1%. 

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2. A nearly-optimal bound for fast regression with ℓ∞ guarantee

   Song, Zhao; Ye, Mingquan; Yin, Junze; Zhang, Lichen (July 2023, Proceedings of Machine Learning Research)

   Given a matrix A ∈ ℝn\texttimes{}d and a vector b ∈ ℝn, we consider the regression problem with ℓ∞ guarantees: finding a vector x′ ∈ ℝd such that $$||x'-x^* ||_infty leq frac{epsilon}{sqrt{d}}cdot ||Ax^*-b||_2cdot ||A^dagger||$$, where x* = arg minx∈Rd ||Ax – b||2. One popular approach for solving such ℓ2 regression problem is via sketching: picking a structured random matrix S ∈ ℝm\texttimes{}n with m < n and S A can be quickly computed, solve the "sketched" regression problem arg minx∈ℝd ||S Ax – Sb||2. In this paper, we show that in order to obtain such ℓ∞ guarantee for ℓ2 regression, one has to use sketching matrices that are dense. To the best of our knowledge, this is the first user case in which dense sketching matrices are necessary. On the algorithmic side, we prove that there exists a distribution of dense sketching matrices with m = ε-2d log3(n/δ) such that solving the sketched regression problem gives the ℓ∞ guarantee, with probability at least 1 – δ. Moreover, the matrix S A can be computed in time O(nd log n). Our row count is nearly-optimal up to logarithmic factors, and significantly improves the result in (Price et al., 2017), in which a superlinear in d rows, m = Ω(ε-2d1+γ) for γ ∈ (0, 1) is required. Moreover, we develop a novel analytical framework for ℓ∞ guarantee regression that utilizes the Oblivious Coordinate-wise Embedding (OCE) property introduced in (Song \& Yu, 2021). Our analysis is much simpler and more general than that of (Price et al., 2017). Leveraging this framework, we extend the ℓ∞ guarantee regression result to dense sketching matrices for computing the fast tensor product of vectors. 

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3. Determinant Maximization via Matroid Intersection Algorithms

   Brown, A; Laddha, A.; Pittu, M.; Singh, M.; Tetali, P. (October 2022, Annual Symposium on Foundations of Computer Science)

   Determinant maximization problem gives a general framework that models problems arising in as diverse fields as statistics [Puk06], convex geometry [Kha96], fair allocations [AGSS16], combinatorics [AGV18], spectral graph theory [NST19a], network design, and random processes [KT12]. In an instance of a determinant maximization problem, we are given a collection of vectors U = {v1, . . . , vn} ⊂ Rd , and a goal is to pick a subset S ⊆ U of given vectors to maximize the determinant of the matrix ∑i∈S vivi^T. Often, the set S of picked vectors must satisfy additional combinatorial constraints such as cardinality constraint (|S| ≤ k) or matroid constraint (S is a basis of a matroid defined on the vectors). In this paper, we give a polynomial-time deterministic algorithm that returns a r O(r)-approximation for any matroid of rank r ≤ d. This improves previous results that give e O(r^2)-approximation algorithms relying on e^O(r)-approximate estimation algorithms [NS16, AG17,AGV18, MNST20] for any r ≤ d. All previous results use convex relaxations and their relationship to stable polynomials and strongly log-concave polynomials. In contrast, our algorithm builds on combinatorial algorithms for matroid intersection, which iteratively improve any solution by finding an alternating negative cycle in the exchange graph defined by the matroids. While the det(.) function is not linear, we show that taking appropriate linear approximations at each iteration suffice to give the improved approximation algorithm. 

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4. Accumulations of Projections—A Unified Framework for Random Sketches in Kernel Ridge Regression

   Chen, Yifan; Yang, Yun. (January 2021, Proceedings of Machine Learning Research)

   null (Ed.)

   Building a sketch of an n-by-n empirical kernel matrix is a common approach to accelerate the computation of many kernel methods. In this paper, we propose a unified framework of constructing sketching methods in kernel ridge regression (KRR), which views the sketching matrix S as an accumulation of m rescaled sub-sampling matrices with independent columns. Our framework incorporates two commonly used sketching methods, sub-sampling sketches (known as the Nyström method) and sub-Gaussian sketches, as special cases with m=1 and m=infinity respectively. Under the new framework, we provide a unified error analysis of sketching approximation and show that our accumulation scheme improves the low accuracy of sub-sampling sketches when certain incoherence characteristic is high, and accelerates the more accurate but computationally heavier sub-Gaussian sketches. By optimally choosing the number m of accumulations, we show that a best trade-off between computational efficiency and statistical accuracy can be achieved. In practice, the sketching method can be as efficiently implemented as the sub-sampling sketches, as only minor extra matrix additions are needed. Our empirical evaluations also demonstrate that the proposed method may attain the accuracy close to sub-Gaussian sketches, while is as efficient as sub-sampling-based sketches. 

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5. Bridging Imaging, Genetics, and Diagnosis in a Coupled Low-Dimensional Framework

   Sayan Ghosal, Qiang Chen (October 2019, Medical Image Computing and Computer Assisted Intervention)

   We propose a joint dictionary learning framework that couples imaging and genetics data in a low dimensional subspace as guided by clinical diagnosis. We use a graph regularization penalty to simultaneously capture inter-regional brain interactions and identify the representative set anatomical basis vectors that span the low dimensional space. We further employ group sparsity to find the representative set of genetic basis vectors that span the same latent space. Finally, the latent projection is used to classify patients versus controls. We have evaluated our model on two task fMRI paradigms and single nucleotide polymorphism (SNP) data from schizophrenic patients and matched neurotypical controls. We employ a ten fold cross validation technique to show the predictive power of our model. We compare our model with canonical correlation analysis of imaging and genetics data and random forest classification. Our approach shows better prediction accuracy on both task datasets. Moreover, the implicated brain regions and genetic variants underlie the well documented deficits in schizophrenia. 

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