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A de Bruijn sequence of order $k$ over a finite alphabet is a cyclic sequence with the property that it contains every possible $k$-sequence as a substring exactly once. Orthogonal de Bruijn sequences are collections of de Bruijn sequences of the same order, $k$, satisfying the joint constraint that every $(k+1)$-sequence appears as a substring in at most one of the sequences in the collection. Both de Bruijn and orthogonal de Bruijn sequences have found numerous applications in synthetic biology, although the latter topic remains largely unexplored in the coding theory literature. Here we study three relevant practical generalizations of orthogonal de Bruijn sequences where we relax either the constraint that every $(k+1)$-sequence appears exactly once, or that the sequences themselves are de Bruijn rather than balanced de Bruijn sequences. We also provide lower and upper bounds on the number of fixed-weight orthogonal de Bruijn sequences., Comment: 7 pages, 3 figures

Data deduplication, one of the key features of modern Big Data storage devices, is the process of removing replicas of data chunks stored by different users. Despite the importance of deduplication, several drawbacks of the method, such as storage robustness and file fragmentation, have not been previously analyzed from a theoretical point of view. Storage robustness pertains to ensuring that deduplicated data can be used to reconstruct the original files without service disruptions and data loss. Fragmentation pertains to the problems of placing deduplicated data chunks of different user files in a proximity-preserving linear order, since neighboring chunks of the same file may be stored in sectors far apart on the server. This work proposes a new theoretical model for data fragmentation and introduces novel graph- and coding-theoretic approaches for reducing fragmentation via limited duplication (repetition coding) and coded deduplication (e.g., linear coding). In addition to alleviating issues with fragmentation, limited duplication and coded deduplication can also serve the dual purpose of increasing the robusteness of the system design. The contributions of our work are three-fold. First, we describe a new model for file structures in the form of self-avoiding (simple) paths in specialized graphs. Second, we introduce several new metrics for measuring the fragmentation level in deduplication systems on graph-structured files, including the stretch metric that captures the worst-case "spread" of adjacent data chunks within a file when deduplicated and placed on the server; and, the jump metric that captures the worst-case number of times during the reconstruction process of a file that one has to change the readout location on the server. For the stretch metric, we establish a connection between the level of fragmentation and the bandwidth of the file-graph. In particular, ...

The performance of Transfer Learning (TL) heavily relies on effective pretraining, which demands large datasets and substantial computational resources. As a result, executing TL is often challenging for individual model developers. Federated Learning (FL) addresses these issues by facilitating collaborations among clients, expanding the dataset indirectly, distributing computational costs, and preserving privacy. However, key challenges remain unresolved. First, existing FL methods tend to optimize transferability only within local domains, neglecting the global learning domain. Second, most approaches rely on indirect transferability metrics, which do not accurately reflect the final target loss or true degree of transferability. To address these gaps, we propose two enhancements to FL. First, we introduce a client-server exchange protocol that leverages cross-client Jacobian (gradient) norms to boost transferability. Second, we increase the average Jacobian norm across clients at the server, using this as a local regularizer to reduce cross-client Jacobian variance. Our transferable federated algorithm, termed FedGTST (Federated Global Transferability via Statistics Tuning), demonstrates that increasing the average Jacobian and reducing its variance allows for tighter control of the target loss. This leads to an upper bound on the target loss in terms of the source loss and source-target domain discrepancy. Extensive experiments on datasets such as MNIST to MNIST-M and CIFAR10 to SVHN show that FedGTST outperforms relevant baselines, including FedSR. On the second dataset pair, FedGTST improves accuracy by 9.8% over FedSR and 7.6% over FedIIR when LeNet is used as the backbone.

Rank aggregation combines multiple ranked lists into a consensus ranking. In fields like biomedical data sharing, rankings may be distributed and require privacy. This motivates the need for federated rank aggregation protocols, which support distributed, private, and communication-efficient learning across multiple clients with local data. We present the first known federated rank aggregation methods using Borda scoring and Lehmer codes, focusing on the sample complexity for federated algorithms on Mallows distributions with a known scaling factor $\phi$ and an unknown centroid permutation $\sigma_0$. Federated Borda approach involves local client scoring, nontrivial quantization, and privacy-preserving protocols. We show that for $\phi \in [0,1)$, and arbitrary $\sigma_0$ of length $N$, it suffices for each of the $L$ clients to locally aggregate $\max\{C_1(\phi), C_2(\phi)\frac{1}{L}\log \frac{N}{\delta}\}$ rankings, where $C_1(\phi)$ and $C_2(\phi)$ are constants, quantize the result, and send it to the server who can then recover $\sigma_0$ with probability $\geq 1-\delta$. Communication complexity scales as $NL \log N$. Our results represent the first rigorous analysis of Borda's method in centralized and distributed settings under the Mallows model. Federated Lehmer coding approach creates a local Lehmer code for each client, using a coordinate-majority aggregation approach with specialized quantization methods for efficiency and privacy. We show that for $\phi+\phi^2<1+\phi^N$, and arbitrary $\sigma_0$ of length $N$, it suffices for each of the $L$ clients to locally aggregate $\max\{C_3(\phi), C_4(\phi)\frac{1}{L}\log \frac{N}{\delta}\}$ rankings, where $C_3(\phi)$ and $C_4(\phi)$ are constants. Clients send truncated Lehmer coordinate histograms to the server, which can recover $\sigma_0$ with probability $\geq 1-\delta$. Communication complexity is $\sim O(N\log NL\log L)$.

A combinatorial trade is a pair of sets of blocks of elements that can be exchanged while preserving relevant subset intersection constraints. The class of balanced and swap-robust minimal trades was proposed in [1] for exchanging blocks of data chunks stored on distributed storage systems in an access- and load-balanced manner. More precisely, data chunks in the trades of interest are labeled by popularity ranks and the blocks are required to have both balanced overall popularity and stability properties with respect to swaps in chunk popularities. The original construction of such trades relied on computer search and paired balanced sets obtained through iterative combining of smaller sets that have provable stability guarantees. To reduce the substantial gap between the results of prior approaches and the known theoretical lower bound, we present new analytical upper and lower bounds on the minimal disbalance of blocks introduced by limited-magnitude popularity ranking swaps. Our constructive and near-optimal approach relies on pairs of graphs whose vertices are two balanced sets with edges/arcs that capture the balance and potential balance changes induced by limited-magnitude popularity swaps. In particular, we show that if we start with carefully selected balanced trades and limit the magnitude of rank swaps to one, the new upper and lower bound on the maximum block disbalance caused by a swap only differ by a factor of $1.07$. We also extend these results for larger popularity swap magnitudes., Comment: arXiv admin note: text overlap with arXiv:2303.12996

Graph neural networks (GNNs) have become instrumental in diverse real-world applications, offering powerful graph learning capabilities for tasks such as social networks and medical data analysis. Despite their successes, GNNs are vulnerable to adversarial attacks, including membership inference attacks (MIA), which threaten privacy by identifying whether a record was part of the model's training data. While existing research has explored MIA in GNNs under graph inductive learning settings, the more common and challenging graph transductive learning setting remains understudied in this context. This paper addresses this gap and proposes an effective two-stage defense, Graph Transductive Defense (GTD), tailored to graph transductive learning characteristics. The gist of our approach is a combination of a train-test alternate training schedule and flattening strategy, which successfully reduces the difference between the training and testing loss distributions. Extensive empirical results demonstrate the superior performance of our method (a decrease in attack AUROC by $9.42\%$ and an increase in utility performance by $18.08\%$ on average compared to LBP), highlighting its potential for seamless integration into various classification models with minimal overhead.

In our problem, we are given access to a number of sequences of nonnegative i.i.d. random variables, whose realizations are observed sequentially. All sequences are of the same finite length. The goal is to pick one element from each sequence in order to maximize a reward equal to the expected value of the sum of the selections from all sequences. The decision on which element to pick is irrevocable, i.e., rejected observations cannot be revisited. Furthermore, the procedure terminates upon having a single selection from each sequence. Our observation constraint is that we cannot observe the current realization of all sequences at each time instant. Instead, we can observe only a smaller, yet arbitrary, subset of them. Thus, together with a stopping rule that determines whether we choose or reject the sample, the solution requires a sampling rule that determines which sequence to observe at each instant. The problem can be solved via dynamic programming, but with an exponential complexity in the length of the sequences. In order to make the solution computationally tractable, we introduce a decoupling approach and determine each stopping time using either a single-sequence dynamic programming, or a Prophet Inequality inspired threshold method, with polynomial complexity in the length of the sequences. We prove that the decoupling approach guarantees at least 0.745 of the optimal expected reward of the joint problem. In addition, we describe how to efficiently compute the optimal number of samples for each sequence, and its' dependence on the variances., Comment: 6 pages, 1 figure

We study the problem of online conditional distribution estimation with \emph{unbounded} label sets under local differential privacy. Let $\mathcal{F}$ be a distribution-valued function class with unbounded label set. We aim at estimating an \emph{unknown} function $f\in \mathcal{F}$ in an online fashion so that at time $t$ when the context $\boldsymbol{x}_t$ is provided we can generate an estimate of $f(\boldsymbol{x}_t)$ under KL-divergence knowing only a privatized version of the true labels sampling from $f(\boldsymbol{x}_t)$. The ultimate objective is to minimize the cumulative KL-risk of a finite horizon $T$. We show that under $(\epsilon,0)$-local differential privacy of the privatized labels, the KL-risk grows as $\tilde{\Theta}(\frac{1}{\epsilon}\sqrt{KT})$ upto poly-logarithmic factors where $K=|\mathcal{F}|$. This is in stark contrast to the $\tilde{\Theta}(\sqrt{T\log K})$ bound demonstrated by Wu et al. (2023a) for bounded label sets. As a byproduct, our results recover a nearly tight upper bound for the hypothesis selection problem of gopi et al. (2020) established only for the batch setting.

Dictionary learning (DL) is commonly used in computational biology to tackle ubiquitous clustering problems due to its conceptual simplicity and relatively low computational complexity. However, DL algorithms produce results that lack interpretability and are not optimized for large-scale graph-structured data. We propose a novel DL algorithm called online convex network dictionary learning (online cvxNDL) that can handle extremely large datasets and enables the interpretation of dictionary elements, which serve as cluster representatives, through convex combinations of real measurements. Moreover, the algorithm can be applied to network-structured data via specialized subnetwork sampling techniques. To demonstrate the utility of our approach, we apply cvxNDL on 3D-genome RNAPII ChIA-Drop data to identify important long-range interaction patterns. ChIA-Drop probes higher-order interactions, and produces hypergraphs whose nodes represent genomic fragments. The hyperedges represent observed physical contacts. Our hypergraph model analysis creates an interpretable dictionary of long-range interaction patterns that accurately represent global chromatin physical contact maps. Using dictionary information, one can also associate the contact maps with RNA transcripts and infer cellular functions. Our results offer two key insights. First, we demonstrate that online cvxNDL retains the accuracy of classical DL methods while simultaneously ensuring unique interpretability and scalability. Second, we identify distinct collections of proximal and distal interaction patterns involving chromatin elements shared by related processes across different chromosomes, as well as patterns unique to specific chromosomes. To associate the dictionary elements with biological properties of the corresponding chromatin regions, we employ Gene Ontology enrichment analysis and perform RNA coexpression studies.

In this review paper, we delve into the nascent field of molecular data storage, focusing on system implementations and code constructions. We start by providing an overview of basic concepts in synthetic and computational biology. Afterwards, we proceed with a review of the diverse approaches followed to implement such systems. In the process, we identify new problems in communication and coding theory, and discuss some relevant results pertaining to DNA sequence profiles, coded trace reconstruction, coding for DNA punchcard systems and coding for unique reconstruction., Comment: Review paper

Hierarchical and tree-like data sets arise in many applications, including language processing, graph data mining, phylogeny and genomics. It is known that tree-like data cannot be embedded into Euclidean spaces of finite dimension with small distortion. This problem can be mitigated through the use of hyperbolic spaces. When such data also has to be processed in a distributed and privatized setting, it becomes necessary to work with new federated learning methods tailored to hyperbolic spaces. As an initial step towards the development of the field of federated learning in hyperbolic spaces, we propose the first known approach to federated classification in hyperbolic spaces. Our contributions are as follows. First, we develop distributed versions of convex SVM classifiers for Poincar\'e discs. In this setting, the information conveyed from clients to the global classifier are convex hulls of clusters present in individual client data. Second, to avoid label switching issues, we introduce a number-theoretic approach for label recovery based on the so-called integer $B_h$ sequences. Third, we compute the complexity of the convex hulls in hyperbolic spaces to assess the extent of data leakage; at the same time, in order to limit communication cost for the hulls, we propose a new quantization method for the Poincar\'e disc coupled with Reed-Solomon-like encoding. Fourth, at the server level, we introduce a new approach for aggregating convex hulls of the clients based on balanced graph partitioning. We test our method on a collection of diverse data sets, including hierarchical single-cell RNA-seq data from different patients distributed across different repositories that have stringent privacy constraints. The classification accuracy of our method is up to $\sim 11\%$ better than its Euclidean counterpart, demonstrating the importance of privacy-preserving learning in hyperbolic spaces., Comment: Published in the Transactions on Machine Learning Research (TMLR). Link: https://openreview.net/forum?id=umggDfMHha

Pathogenic infections pose a significant threat to global health, affecting millions of people every year and presenting substantial challenges to healthcare systems worldwide. Efficient and timely testing plays a critical role in disease control and transmission prevention. Group testing is a well-established method for reducing the number of tests needed to screen large populations when the disease prevalence is low. However, it does not fully utilize the quantitative information provided by qPCR methods, nor is it able to accommodate a wide range of pathogen loads. To address these issues, we introduce a novel adaptive semi-quantitative group testing (SQGT) scheme to efficiently screen populations via two-stage qPCR testing. The SQGT method quantizes cycle threshold ($Ct$) values into multiple bins, leveraging the information from the first stage of screening to improve the detection sensitivity. Dynamic $Ct$ threshold adjustments mitigate dilution effects and enhance test accuracy. Comparisons with traditional binary outcome GT methods show that SQGT reduces the number of tests by $24$% while maintaining a negligible false negative rate., Comment: Corrected a misspelled name in the author list on page 1

GNNs can inadvertently expose sensitive user information and interactions through their model predictions. To address these privacy concerns, Differential Privacy (DP) protocols are employed to control the trade-off between provable privacy protection and model utility. Applying standard DP approaches to GNNs directly is not advisable due to two main reasons. First, the prediction of node labels, which relies on neighboring node attributes through graph convolutions, can lead to privacy leakage. Second, in practical applications, the privacy requirements for node attributes and graph topology may differ. In the latter setting, existing DP-GNN models fail to provide multigranular trade-offs between graph topology privacy, node attribute privacy, and GNN utility. To address both limitations, we propose a new framework termed Graph Differential Privacy (GDP), specifically tailored to graph learning. GDP ensures both provably private model parameters as well as private predictions. Additionally, we describe a novel unified notion of graph dataset adjacency to analyze the properties of GDP for different levels of graph topology privacy. Our findings reveal that DP-GNNs, which rely on graph convolutions, not only fail to meet the requirements for multigranular graph topology privacy but also necessitate the injection of DP noise that scales at least linearly with the maximum node degree. In contrast, our proposed Differentially Private Decoupled Graph Convolutions (DPDGCs) represent a more flexible and efficient alternative to graph convolutions that still provides the necessary guarantees of GDP. To validate our approach, we conducted extensive experiments on seven node classification benchmarking and illustrative synthetic datasets. The results demonstrate that DPDGCs significantly outperform existing DP-GNNs in terms of privacy-utility trade-offs., Comment: NeurIPS 2023

The eXtreme Multi-label Classification~(XMC) problem seeks to find relevant labels from an exceptionally large label space. Most of the existing XMC learners focus on the extraction of semantic features from input query text. However, conventional XMC studies usually neglect the side information of instances and labels, which can be of use in many real-world applications such as recommendation systems and e-commerce product search. We propose Predicted Instance Neighborhood Aggregation (PINA), a data enhancement method for the general XMC problem that leverages beneficial side information. Unlike most existing XMC frameworks that treat labels and input instances as featureless indicators and independent entries, PINA extracts information from the label metadata and the correlations among training instances. Extensive experimental results demonstrate the consistent gain of PINA on various XMC tasks compared to the state-of-the-art methods: PINA offers a gain in accuracy compared to standard XR-Transformers on five public benchmark datasets. Moreover, PINA achieves a $\sim 5\%$ gain in accuracy on the largest dataset LF-AmazonTitles-1.3M. Our implementation is publicly available., Comment: ICML 2023

Motivated by testing for pathogenic diseases we consider a new nonadaptive group testing problem for which: (1) positives occur within a burst, capturing the fact that infected test subjects often come in clusters, and (2) that the test outcomes arise from semiquantitative measurements that provide coarse information about the number of positives in any tested group. Our model generalizes prior work on detecting a single burst of positives with classical group testing[1] as well as work on semiquantitative group testing (SQGT)[2]. Specifically, we study the setting where the burst-length $\ell$ is known and the semiquantitative tests provide potentially nonuniform estimates on the number of positives in a test group. The estimates represent the index of a quantization bin containing the (exact) total number of positives, for arbitrary thresholds $\eta_1,\dots,\eta_s$. Interestingly, we show that the minimum number of tests needed for burst identification is essentially only a function of the largest threshold $\eta_s$. In this context, our main result is an order-optimal test scheme that can recover any burst of length $\ell$ using roughly $\frac{\ell}{2\eta_s}+\log_{s+1}(n)$ measurements. This suggests that a large saturation level $\eta_s$ is more important than finely quantized information when dealing with bursts. We also provide results for related modeling assumptions and specialized choices of thresholds.

Balanced and swap-robust minimal trades, introduced in [1], are important for studying the balance and stability of server access request protocols under data popularity changes. Constructions of such trades have so far relied on paired sets obtained through iterative combining of smaller sets that have provable stability guarantees, coupled with exhaustive computer search. Currently, there exists a nonnegligible gap between the resulting total dynamic balance discrepancy and the known theoretical lower bound. We present both new upper and lower bounds on the total service requests discrepancy under limited popularity changes. Our constructive near-optimal approach uses a new class of paired graphs whose vertices are two balanced sets with edges (arcs) that capture the balance and potential balance changes induced by limited-magnitude popularity changes (swaps).

Motivated by applications in polymer-based data storage we introduced the new problem of characterizing the code rate and designing constant-weight binary $B_2$-sequences. Binary $B_2$-sequences are collections of binary strings of length $n$ with the property that the real-valued sums of all distinct pairs of strings are distinct. In addition to this defining property, constant-weight binary $B_2$-sequences also satisfy the constraint that each string has a fixed, relatively small weight $\omega$ that scales linearly with $n$. The constant-weight constraint ensures low-cost synthesis and uniform processing of the data readout via tandem mass spectrometers. Our main results include upper bounds on the size of the codes formulated as entropy-optimization problems and constructive lower bounds based on Sidon sequences.

The Secretary problem is a classical sequential decision-making question that can be succinctly described as follows: a set of rank-ordered applicants are interviewed sequentially for a single position. Once an applicant is interviewed, an immediate and irrevocable decision is made if the person is to be offered the job or not and only applicants observed so far can be used in the decision process. The problem of interest is to identify the stopping rule that maximizes the probability of hiring the highest-ranked applicant. A multiple-choice version of the Secretary problem, known as the Dowry problem, assumes that one is given a fixed integer budget for the total number of selections allowed to choose the best applicant. It has been solved using tools from dynamic programming and optimal stopping theory. We provide the first combinatorial proof for a related new \emph{query-based model} for which we are allowed to solicit the response of an expert to determine if an applicant is optimal. Since the selection criteria differ from those of the Dowry problem we obtain nonidentical expected stopping times. Our result indicates that an optimal strategy is the $(a_s, a_{s-1}, \ldots, a_1)$-strategy, i.e., for the $i^{th}$ selection, where $1 \le i \le s$ and $1 \le j = s+1-i \le s$, we reject the first $a_j$ applicants, wait until the decision of the $(i-1)^{th}$ selection (if $i \ge 2$), and then accept the next applicant whose qualification is better than all previously appeared applicants. Furthermore, our optimal strategy is right-hand based, i.e., the optimal strategies for two models with $s_1$ and $s_2$ selections in total ($s_1 < s_2$) share the same sequence $a_1, a_2, \ldots, a_{s_1}$ when it is viewed from the right. When the total number of applicants tends to infinity, our result agrees with the thresholds obtained by Gilbert and Mosteller., Comment: 15 pages, 1 figure, 1 table. arXiv admin note: substantial text overlap with arXiv:2101.07250

As the demand for user privacy grows, controlled data removal (machine unlearning) is becoming an important feature of machine learning models for data-sensitive Web applications such as social networks and recommender systems. Nevertheless, at this point it is still largely unknown how to perform efficient machine unlearning of graph neural networks (GNNs); this is especially the case when the number of training samples is small, in which case unlearning can seriously compromise the performance of the model. To address this issue, we initiate the study of unlearning the Graph Scattering Transform (GST), a mathematical framework that is efficient, provably stable under feature or graph topology perturbations, and offers graph classification performance comparable to that of GNNs. Our main contribution is the first known nonlinear approximate graph unlearning method based on GSTs. Our second contribution is a theoretical analysis of the computational complexity of the proposed unlearning mechanism, which is hard to replicate for deep neural networks. Our third contribution are extensive simulation results which show that, compared to complete retraining of GNNs after each removal request, the new GST-based approach offers, on average, a 10.38x speed-up and leads to a 2.6% increase in test accuracy during unlearning of 90 out of 100 training graphs from the IMDB dataset (10% training ratio). Our implementation is available online at https://doi.org/10.5281/zenodo.7613150., Comment: 12 pages. WWW 2023

Federated clustering (FC) is an unsupervised learning problem that arises in a number of practical applications, including personalized recommender and healthcare systems. With the adoption of recent laws ensuring the "right to be forgotten", the problem of machine unlearning for FC methods has become of significant importance. We introduce, for the first time, the problem of machine unlearning for FC, and propose an efficient unlearning mechanism for a customized secure FC framework. Our FC framework utilizes special initialization procedures that we show are well-suited for unlearning. To protect client data privacy, we develop the secure compressed multiset aggregation (SCMA) framework that addresses sparse secure federated learning (FL) problems encountered during clustering as well as more general problems. To simultaneously facilitate low communication complexity and secret sharing protocols, we integrate Reed-Solomon encoding with special evaluation points into our SCMA pipeline, and prove that the client communication cost is logarithmic in the vector dimension. Additionally, to demonstrate the benefits of our unlearning mechanism over complete retraining, we provide a theoretical analysis for the unlearning performance of our approach. Simulation results show that the new FC framework exhibits superior clustering performance compared to previously reported FC baselines when the cluster sizes are highly imbalanced. Compared to completely retraining K-means++ locally and globally for each removal request, our unlearning procedure offers an average speed-up of roughly 84x across seven datasets. Our implementation for the proposed method is available at https://github.com/thupchnsky/mufc., Comment: 27 pages. ICLR 2023

In 2021, the Coordinated Science Laboratory CSL, an Interdisciplinary Research Unit at the University of Illinois Urbana-Champaign, hosted the Future of Computing Symposium to celebrate its 70th anniversary. CSL's research covers the full computing stack, computing's impact on society and the resulting need for social responsibility. In this white paper, we summarize the major technological points, insights, and directions that speakers brought forward during the Future of Computing Symposium. Participants discussed topics related to new computing paradigms, technologies, algorithms, behaviors, and research challenges to be expected in the future. The symposium focused on new computing paradigms that are going beyond traditional computing and the research needed to support their realization. These needs included stressing security and privacy, the end to end human cyber physical systems and with them the analysis of the end to end artificial intelligence needs. Furthermore, advances that enable immersive environments for users, the boundaries between humans and machines will blur and become seamless. Particular integration challenges were made clear in the final discussion on the integration of autonomous driving, robo taxis, pedestrians, and future cities. Innovative approaches were outlined to motivate the next generation of researchers to work on these challenges. The discussion brought out the importance of considering not just individual research areas, but innovations at the intersections between computing research efforts and relevant application domains, such as health care, transportation, energy systems, and manufacturing.

Graph-structured data is ubiquitous in practice and often processed using graph neural networks (GNNs). With the adoption of recent laws ensuring the ``right to be forgotten'', the problem of graph data removal has become of significant importance. To address the problem, we introduce the first known framework for \emph{certified graph unlearning} of GNNs. In contrast to standard machine unlearning, new analytical and heuristic unlearning challenges arise when dealing with complex graph data. First, three different types of unlearning requests need to be considered, including node feature, edge and node unlearning. Second, to establish provable performance guarantees, one needs to address challenges associated with feature mixing during propagation. The underlying analysis is illustrated on the example of simple graph convolutions (SGC) and their generalized PageRank (GPR) extensions, thereby laying the theoretical foundation for certified unlearning of GNNs. Our empirical studies on six benchmark datasets demonstrate excellent performance-complexity trade-offs when compared to complete retraining methods and approaches that do not leverage graph information. For example, when unlearning $20\%$ of the nodes on the Cora dataset, our approach suffers only a $0.1\%$ loss in test accuracy while offering a $4$-fold speed-up compared to complete retraining. Our scheme also outperforms unlearning methods that do not leverage graph information with a $12\%$ increase in test accuracy for a comparable time complexity., Comment: NeurIPS 2022 New Frontiers in Graph Learning Workshop (NeurIPS GLFrontiers 2022)

The problem of fitting distances by tree-metrics has received significant attention in the theoretical computer science and machine learning communities alike, due to many applications in natural language processing, phylogeny, cancer genomics and a myriad of problem areas that involve hierarchical clustering. Despite the existence of several provably exact algorithms for tree-metric fitting of data that inherently obeys tree-metric constraints, much less is known about how to best fit tree-metrics for data whose structure moderately (or substantially) differs from a tree. For such noisy data, most available algorithms perform poorly and often produce negative edge weights in representative trees. Furthermore, it is currently not known how to choose the most suitable approximation objective for noisy fitting. Our contributions are as follows. First, we propose a new approach to tree-metric denoising (HyperAid) in hyperbolic spaces which transforms the original data into data that is ``more'' tree-like, when evaluated in terms of Gromov's $\delta$ hyperbolicity. Second, we perform an ablation study involving two choices for the approximation objective, $\ell_p$ norms and the Dasgupta loss. Third, we integrate HyperAid with schemes for enforcing nonnegative edge-weights. As a result, the HyperAid platform outperforms all other existing methods in the literature, including Neighbor Joining (NJ), TreeRep and T-REX, both on synthetic and real-world data. Synthetic data is represented by edge-augmented trees and shortest-distance metrics while the real-world datasets include Zoo, Iris, Glass, Segmentation and SpamBase; on these datasets, the average improvement with respect to NJ is $125.94\%$., Comment: To appear in KDD 2022

Many high-dimensional practical data sets have hierarchical structures induced by graphs or time series. Such data sets are hard to process in Euclidean spaces and one often seeks low-dimensional embeddings in other space forms to perform the required learning tasks. For hierarchical data, the space of choice is a hyperbolic space because it guarantees low-distortion embeddings for tree-like structures. The geometry of hyperbolic spaces has properties not encountered in Euclidean spaces that pose challenges when trying to rigorously analyze algorithmic solutions. We propose a unified framework for learning scalable and simple hyperbolic linear classifiers with provable performance guarantees. The gist of our approach is to focus on Poincar\'e ball models and formulate the classification problems using tangent space formalisms. Our results include a new hyperbolic perceptron algorithm as well as an efficient and highly accurate convex optimization setup for hyperbolic support vector machine classifiers. Furthermore, we adapt our approach to accommodate second-order perceptrons, where data is preprocessed based on second-order information (correlation) to accelerate convergence, and strategic perceptrons, where potentially manipulated data arrives in an online manner and decisions are made sequentially. The excellent performance of the Poincar\'e second-order and strategic perceptrons shows that the proposed framework can be extended to general machine learning problems in hyperbolic spaces. Our experimental results, pertaining to synthetic, single-cell RNA-seq expression measurements, CIFAR10, Fashion-MNIST and mini-ImageNet, establish that all algorithms provably converge and have complexity comparable to those of their Euclidean counterparts. Accompanying codes can be found at: https://github.com/thupchnsky/PoincareLinearClassification., Comment: arXiv admin note: substantial text overlap with arXiv:2109.03781

The $k$-deck problem is concerned with finding the smallest positive integer $S(k)$ such that there exist at least two strings of length $S(k)$ that share the same $k$-deck, i.e., the multiset of subsequences of length $k$. We introduce the new problem of gapped $k$-deck reconstruction: For a given gap parameter $s$, we seek the smallest positive integer $G_s(k)$ such that there exist at least two distinct strings of length $G_s(k)$ that cannot be distinguished based on a "gapped" set of $k$-subsequences. The gap constraint requires the elements in the subsequences to be at least $s$ positions apart within the original string. Our results are as follows. First, we show how to construct sequences sharing the same $2$-gapped $k$-deck using a nontrivial modification of the recursive Morse-Thue string construction procedure. This establishes the first known constructive upper bound on $G_2(k)$. Second, we further improve this bound using the approach by Dudik and Schulman.

Trades, introduced by Hedayat, are two sets of blocks of elements which may be exchanged (traded) without altering the counts of certain subcollections of elements within their constituent blocks. They are of importance in applications where certain combinations of elements dynamically become prohibited from being placed in the same group of elements, since in this case one can trade the offending blocks with allowed ones. This is particularly the case in distributed storage systems, where due to privacy and other constraints, data of some groups of users cannot be stored together on the same server. We introduce a new class of balanced trades, important for access balancing of servers, and perturbation resilient balanced trades, important for studying the stability of server access frequencies with respect to changes in data popularity. The constructions and bounds on our new trade schemes rely on specialized selections of defining sets in minimal trades and number-theoretic analyses., Comment: 6 pages

Learning on graphs has attracted significant attention in the learning community due to numerous real-world applications. In particular, graph neural networks (GNNs), which take numerical node features and graph structure as inputs, have been shown to achieve state-of-the-art performance on various graph-related learning tasks. Recent works exploring the correlation between numerical node features and graph structure via self-supervised learning have paved the way for further performance improvements of GNNs. However, methods used for extracting numerical node features from raw data are still graph-agnostic within standard GNN pipelines. This practice is sub-optimal as it prevents one from fully utilizing potential correlations between graph topology and node attributes. To mitigate this issue, we propose a new self-supervised learning framework, Graph Information Aided Node feature exTraction (GIANT). GIANT makes use of the eXtreme Multi-label Classification (XMC) formalism, which is crucial for fine-tuning the language model based on graph information, and scales to large datasets. We also provide a theoretical analysis that justifies the use of XMC over link prediction and motivates integrating XR-Transformers, a powerful method for solving XMC problems, into the GIANT framework. We demonstrate the superior performance of GIANT over the standard GNN pipeline on Open Graph Benchmark datasets: For example, we improve the accuracy of the top-ranked method GAMLP from $68.25\%$ to $69.67\%$, SGC from $63.29\%$ to $66.10\%$ and MLP from $47.24\%$ to $61.10\%$ on the ogbn-papers100M dataset by leveraging GIANT., Comment: Published in ICLR 2022

The problem of reconstructing strings from substring information has found many applications due to its importance in genomic data sequencing and DNA- and polymer-based data storage. One practically important and challenging paradigm requires reconstructing mixtures of strings based on the union of compositions of their prefixes and suffixes, generated by mass spectrometry devices. We describe new coding methods that allow for unique joint reconstruction of subsets of strings selected from a code and provide upper and lower bounds on the asymptotic rate of the underlying codebooks. Our code constructions combine properties of binary Bh and Dyck strings and that can be extended to accommodate missing substrings in the pool. As auxiliary results, we obtain the first known bounds on binary Bh sequences for arbitrary even parameters h, and also describe various error models inherent to mass spectrometry analysis. This paper contains a correction of the prior work by the authors, published in [24]. In particular, the bounds on the prefix codes are now corrected.

Many high-dimensional and large-volume data sets of practical relevance have hierarchical structures induced by trees, graphs or time series. Such data sets are hard to process in Euclidean spaces and one often seeks low-dimensional embeddings in other space forms to perform required learning tasks. For hierarchical data, the space of choice is a hyperbolic space since it guarantees low-distortion embeddings for tree-like structures. Unfortunately, the geometry of hyperbolic spaces has properties not encountered in Euclidean spaces that pose challenges when trying to rigorously analyze algorithmic solutions. Here, for the first time, we establish a unified framework for learning scalable and simple hyperbolic linear classifiers with provable performance guarantees. The gist of our approach is to focus on Poincar\'e ball models and formulate the classification problems using tangent space formalisms. Our results include a new hyperbolic and second-order perceptron algorithm as well as an efficient and highly accurate convex optimization setup for hyperbolic support vector machine classifiers. All algorithms provably converge and are highly scalable as they have complexities comparable to those of their Euclidean counterparts. Their performance accuracies on synthetic data sets comprising millions of points, as well as on complex real-world data sets such as single-cell RNA-seq expression measurements, CIFAR10, Fashion-MNIST and mini-ImageNet., Comment: A short version of this paper appears in ICDM 2021

Candidates arrive sequentially for an interview process which results in them being ranked relative to their predecessors. Based on the ranks available at each time, one must develop a decision mechanism that selects or dismisses the current candidate in an effort to maximize the chance to select the best. This classical version of the ``Secretary problem'' has been studied in depth using mostly combinatorial approaches, along with numerous other variants. In this work we consider a particular new version where during reviewing one is allowed to query an external expert to improve the probability of making the correct decision. Unlike existing formulations, we consider experts that are not necessarily infallible and may provide suggestions that can be faulty. For the solution of our problem we adopt a probabilistic methodology and view the querying times as consecutive stopping times which we optimize with the help of optimal stopping theory. For each querying time we must also design a mechanism to decide whether we should terminate the search at the querying time or not. This decision is straightforward under the usual assumption of infallible experts but, when experts are faulty, it has a far more intricate structure., Comment: 24 pages, 1 figure, 1 table

Hypergraphs are used to model higher-order interactions amongst agents and there exist many practically relevant instances of hypergraph datasets. To enable efficient processing of hypergraph-structured data, several hypergraph neural network platforms have been proposed for learning hypergraph properties and structure, with a special focus on node classification. However, almost all existing methods use heuristic propagation rules and offer suboptimal performance on many datasets. We propose AllSet, a new hypergraph neural network paradigm that represents a highly general framework for (hyper)graph neural networks and for the first time implements hypergraph neural network layers as compositions of two multiset functions that can be efficiently learned for each task and each dataset. Furthermore, AllSet draws on new connections between hypergraph neural networks and recent advances in deep learning of multiset functions. In particular, the proposed architecture utilizes Deep Sets and Set Transformer architectures that allow for significant modeling flexibility and offer high expressive power. To evaluate the performance of AllSet, we conduct the most extensive experiments to date involving ten known benchmarking datasets and three newly curated datasets that represent significant challenges for hypergraph node classification. The results demonstrate that AllSet has the unique ability to consistently either match or outperform all other hypergraph neural networks across the tested datasets., Comment: ICLR 2022

Embedding methods for product spaces are powerful techniques for low-distortion and low-dimensional representation of complex data structures. Here, we address the new problem of linear classification in product space forms -- products of Euclidean, spherical, and hyperbolic spaces. First, we describe novel formulations for linear classifiers on a Riemannian manifold using geodesics and Riemannian metrics which generalize straight lines and inner products in vector spaces. Second, we prove that linear classifiers in $d$-dimensional space forms of any curvature have the same expressive power, i.e., they can shatter exactly $d+1$ points. Third, we formalize linear classifiers in product space forms, describe the first known perceptron and support vector machine classifiers for such spaces and establish rigorous convergence results for perceptrons. Moreover, we prove that the Vapnik-Chervonenkis dimension of linear classifiers in a product space form of dimension $d$ is \emph{at least} $d+1$. We support our theoretical findings with simulations on several datasets, including synthetic data, image data, and single-cell RNA sequencing (scRNA-seq) data. The results show that classification in low-dimensional product space forms for scRNA-seq data offers, on average, a performance improvement of $\sim15\%$ when compared to that in Euclidean spaces of the same dimension.

Semiquantitative group testing (SQGT) is a pooling method in which the test outcomes represent bounded intervals for the number of defectives. Alternatively, it may be viewed as an adder channel with quantized outputs. SQGT represents a natural choice for Covid-19 group testing as it allows for a straightforward interpretation of the cycle threshold values produced by polymerase chain reactions (PCR). Prior work on SQGT did not address the need for adaptive testing with a small number of rounds as required in practice. We propose conceptually simple methods for 2-round and nonadaptive SQGT that significantly improve upon existing schemes by using ideas on nonbinary measurement matrices based on expander graphs and list-disjunct matrices.

We study the secretary problem in which rank-ordered lists are generated by the Mallows model and the goal is to identify the highest-ranked candidate through a sequential interview process which does not allow rejected candidates to be revisited. The main difference between our formulation and existing models is that, during the selection process, we are given a fixed number of opportunities to query an infallible expert whether the current candidate is the highest-ranked or not. If the response is positive, the selection process terminates, otherwise, the search continues until a new potentially optimal candidate is identified. Our optimal interview strategy, as well as the expected number of candidates interviewed and the expected number of queries used, can be determined through the evaluation of well-defined recurrence relations. Specifically, if we are allowed to query $s-1$ times and to make a final selection without querying (thus, making $s$ selections in total) then the optimum scheme is characterized by $s$ thresholds that depend on the parameter $\theta$ of the Mallows distribution but are independent on the maximum number of queries., Comment: 28 pages, 2 figures, 2 tables

The well-known secretary problem in sequential analysis and optimal stopping theory asks one to maximize the probability of finding the optimal candidate in a sequentially examined list under the constraint that accept/reject decisions are made in real-time. A version of the problem is the so-called postdoc problem, for which the question of interest is to devise a strategy that identifies the second-best candidate with highest possible probability of success. We study the postdoc problem in its combinatorial form. In this setting, a permutation $\pi$ of length $N$ is sampled according to some distribution on the symmetric group $S_N$ and the elements of $\pi$ are revealed one-by-one from left to right so that at each step, one can only observe the relative orders of the elements. At each step, one must decide to either accept or reject the currently presented element and cannot recall the decision in the future. The question of interest is to find the optimal strategy for selecting the position of the second-largest value. We solve the postdoc problem for the untraditional setting where the candidates are not presented uniformly at random but rather according to permutations drawn from the Mallows distribution. The Mallows distribution assigns to each permutation $\pi \in S_N$ a weight $\theta^{c(\pi)}$, where the function c counts the number of inversions in $\pi$. To identify the optimal stopping criteria for the significantly more challenging postdoc problem, we adopt a combinatorial methodology that includes new proof techniques and novel methodological extensions compared to the analysis first introduced in the setting of the secretary problem. The optimal strategies depend on the parameter $\theta$ of the Mallows distribution and can be determined exactly by solving well-defined recurrence relations., Comment: 33 pages, 7 figures, 2 tables

The first part of the paper presents a review of the gold-standard testing protocol for Covid-19, real-time, reverse transcriptase PCR, and its properties and associated measurement data such as amplification curves that can guide the development of appropriate and accurate adaptive group testing protocols. The second part of the paper is concerned with examining various off-the-shelf group testing methods for Covid-19 and identifying their strengths and weaknesses for the application at hand. The third part of the paper contains a collection of new analytical results for adaptive semiquantitative group testing with probabilistic and combinatorial priors, including performance bounds, algorithmic solutions, and noisy testing protocols. The probabilistic setting is of special importance as it is designed to be simple to implement by nonexperts and handle heavy hitters. The worst-case paradigm extends and improves upon prior work on semiquantitative group testing with and without specialized PCR noise models.

The problem of string reconstruction from substring information has found many applications due to its relevance in DNA- and polymer-based data storage. One practically important and challenging paradigm requires reconstructing mixtures of strings based on the union of compositions of their prefixes and suffixes, generated by mass spectrometry readouts. We describe new coding methods that allow for unique joint reconstruction of subsets of strings selected from a code and provide matching upper and lower bounds on the asymptotic rate of the underlying codebooks. Under certain mild constraints on the problem parameters, one can show that the largest possible rate of a codebook that allows for all subcollections of $\leq h$ codestrings to be uniquely reconstructable from the prefix-suffix information equals $1/h$.

We propose new repair schemes for Reed-Solomon codes that use subspace polynomials and hence generalize previous works in the literature that employ trace polynomials. The Reed-Solomon codes are over $\mathbb{F}_{q^\ell}$ and have redundancy $r = n-k \geq q^m$, $1\leq m\leq \ell$, where $n$ and $k$ are the code length and dimension, respectively. In particular, for one erasure, we show that our schemes can achieve optimal repair bandwidths whenever $n=q^\ell$ and $r = q^m,$ for all $1 \leq m \leq \ell$. For two erasures, our schemes use the same bandwidth per erasure as the single erasure schemes, for $\ell/m$ is a power of $q$, and for $\ell=q^a$, $m=q^b-1>1$ ($a \geq b \geq 1$), and for $m\geq \ell/2$ when $\ell$ is even and $q$ is a power of two.

Many data analysis problems can be cast as distance geometry problems in \emph{space forms} -- Euclidean, spherical, or hyperbolic spaces. Often, absolute distance measurements are often unreliable or simply unavailable and only proxies to absolute distances in the form of similarities are available. Hence we ask the following: Given only \emph{comparisons} of similarities amongst a set of entities, what can be said about the geometry of the underlying space form? To study this question, we introduce the notions of the \textit{ordinal capacity} of a target space form and \emph{ordinal spread} of the similarity measurements. The latter is an indicator of complex patterns in the measurements, while the former quantifies the capacity of a space form to accommodate a set of measurements with a specific ordinal spread profile. We prove that the ordinal capacity of a space form is related to its dimension and the sign of its curvature. This leads to a lower bound on the Euclidean and spherical embedding dimension of what we term similarity graphs. More importantly, we show that the statistical behavior of the ordinal spread random variables defined on a similarity graph can be used to identify its underlying space form. We support our theoretical claims with experiments on weighted trees, single-cell RNA expression data and spherical cartographic measurements.

In many important graph data processing applications the acquired information includes both node features and observations of the graph topology. Graph neural networks (GNNs) are designed to exploit both sources of evidence but they do not optimally trade-off their utility and integrate them in a manner that is also universal. Here, universality refers to independence on homophily or heterophily graph assumptions. We address these issues by introducing a new Generalized PageRank (GPR) GNN architecture that adaptively learns the GPR weights so as to jointly optimize node feature and topological information extraction, regardless of the extent to which the node labels are homophilic or heterophilic. Learned GPR weights automatically adjust to the node label pattern, irrelevant on the type of initialization, and thereby guarantee excellent learning performance for label patterns that are usually hard to handle. Furthermore, they allow one to avoid feature over-smoothing, a process which renders feature information nondiscriminative, without requiring the network to be shallow. Our accompanying theoretical analysis of the GPR-GNN method is facilitated by novel synthetic benchmark datasets generated by the so-called contextual stochastic block model. We also compare the performance of our GNN architecture with that of several state-of-the-art GNNs on the problem of node-classification, using well-known benchmark homophilic and heterophilic datasets. The results demonstrate that GPR-GNN offers significant performance improvement compared to existing techniques on both synthetic and benchmark data., Comment: ICLR 2021

The problem of estimating the support of a distribution is of great importance in many areas of machine learning, computer science, physics and biology. Most of the existing work in this domain has focused on settings that assume perfectly accurate sampling approaches, which is seldom true in practical data science. Here we introduce the first known approach to support estimation in the presence of sampling artifacts and errors where each sample is assumed to arise from a Poisson repeat channel which simultaneously captures repetitions and deletions of samples. The proposed estimator is based on regularized weighted Chebyshev approximations, with weights governed by evaluations of so-called Touchard (Bell) polynomials. The supports in the presence of sampling artifacts are calculated using discretized semi-infite programming methods. The estimation approach is tested on synthetic and textual data, as well as on GISAID data collected to address a new problem in computational biology: mutational support estimation in genes of the SARS-Cov-2 virus. In the later setting, the Poisson channel captures the fact that many individuals are tested multiple times for the presence of viral RNA, thereby leading to repeated samples, while other individual's results are not recorded due to test errors. For all experiments performed, we observed significant improvements of our integrated methods compared to those obtained through adequate modifications of state-of-the-art noiseless support estimation methods.

Motivated by polymer-based data-storage platforms that use chains of binary synthetic polymers as the recording media and read the content via tandem mass spectrometers, we propose a new family of codes that allows for both unique string reconstruction and correction of multiple mass errors. We consider two approaches: The first approach pertains to asymmetric errors and it is based on introducing redundancy that scales linearly with the number of errors and logarithmically with the length of the string. The construction allows for the string to be uniquely reconstructed based only on its erroneous substring composition multiset. The key idea behind our unique reconstruction approach is to interleave (shifted) Catalan-Bertrand paths with arbitrary binary strings and "reflect" them so as to force prefixes and suffixes of the same length to have different weights. The asymptotic code rate of the scheme is one, and decoding is accomplished via a simplified version of the backtracking algorithm used for the Turnpike problem. For symmetric errors, we use a polynomial characterization of the mass information and adapt polynomial evaluation code constructions for this setting. In the process, we develop new efficient decoding algorithms for a constant number of composition errors and show that the redundancy of the scheme scales quadratically with the number of errors and logarithmically with the codelength., Comment: arXiv admin note: substantial text overlap with arXiv:1904.09280, arXiv:2001.04967

We consider the problem of correcting mass readout errors in information encoded in binary polymer strings. Our work builds on results for string reconstruction problems using composition multisets [Acharya et al., 2015] and the unique string reconstruction framework proposed in [Pattabiraman et al., 2019]. Binary polymer-based data storage systems [Laure et al., 2016] operate by designing two molecules of significantly different masses to represent the symbols $\{{0,1\}}$ and perform readouts through noisy tandem mass spectrometry. Tandem mass spectrometers fragment the strings to be read into shorter substrings and only report their masses, often with errors due to imprecise ionization. Modeling the fragmentation process output in terms of composition multisets allows for designing asymptotically optimal codes capable of unique reconstruction and the correction of a single mass error [Pattabiraman et al., 2019] through the use of derivatives of Catalan paths. Nevertheless, no solutions for multiple-mass error-corrections are currently known. Our work addresses this issue by describing the first multiple-error correction codes that use the polynomial factorization approach for the Turnpike problem [Skiena et al., 1990] and the related factorization described in [Acharya et al., 2015]. Adding Reed-Solomon type coding redundancy into the corresponding polynomials allows for correcting $t$ mass errors in polynomial time using $t^2\, \log\,k$ redundant bits, where $k$ is the information string length. The redundancy can be improved to $\log\,k + t$. However, no decoding algorithm that runs polynomial-time in both $t$ and $n$ for this scheme are currently known, where $n$ is the length of the coded string.

Motivated by applications in topological DNA-based data storage, we introduce and study a novel setting of Non-Adaptive Group Testing (NAGT) with runlength constraints on the columns of the test matrix, in the sense that any two 1's must be separated by a run of at least d 0's. We describe and analyze a probabilistic construction of a runlength-constrained scheme in the zero-error and vanishing error settings, and show that the number of tests required by this construction is optimal up to logarithmic factors in the runlength constraint d and the number of defectives k in both cases. Surprisingly, our results show that runlength-constrained NAGT is not more demanding than unconstrained NAGT when d=O(k), and that for almost all choices of d and k it is not more demanding than NAGT with a column Hamming weight constraint only. Towards obtaining runlength-constrained Quantitative NAGT (QNAGT) schemes with good parameters, we also provide lower bounds for this setting and a nearly optimal probabilistic construction of a QNAGT scheme with a column Hamming weight constraint.

Generative graph models create instances of graphs that mimic the properties of real-world networks. Generative models are successful at retaining pairwise associations in the underlying networks but often fail to capture higher-order connectivity patterns known as network motifs. Different types of graphs contain different network motifs, an example of which are triangles that often arise in social and biological networks. It is hence vital to capture these higher-order structures to simulate real-world networks accurately. We propose Multi-MotifGAN (MMGAN), a motif-targeted Generative Adversarial Network (GAN) that generalizes the benchmark NetGAN approach. The generalization consists of combining multiple biased random walks, each of which captures a different motif structure. MMGAN outperforms NetGAN at creating new graphs that accurately reflect the network motif statistics of input graphs such as Citeseer, Cora and Facebook.