In interactive coding, Alice and Bob wish to compute some function $f$ of their individual private inputs $x$ and $y$. They do this by engaging in an interactive protocol to jointly compute $f(x,y)$. The goal is to do this in an error-resilient way, such that even given some fraction of adversarial corruptions to the protocol, both parties still learn $f(x,y)$. Typically, the error resilient protocols constructed by interactive coding schemes are \emph{non-adaptive}, that is, the length of the protocol as well as the speaker in each round is fixed beforehand. The maximal error resilience obtainable by non-adaptive schemes is now well understood. In order to circumvent known barriers and achieve higher error resilience, the work of Agrawal, Gelles, and Sahai (ISIT 2016) introduced to interactive coding the notion of \emph{adaptive} schemes, where the length of the protocol or the speaker order are no longer necessarily fixed. In this paper, we study the power of \emph{adaptive termination} in the context of the error resilience of interactive coding schemes. In other words, what is the power of schemes where Alice and Bob are allowed to disengage from the protocol early? We study this question in two contexts, both for the task of \emph{message exchange}, where the goal is to learn the other party's input.
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Given some binary matrix $M$, suppose we are presented with the collection of its rows and columns in independent arbitrary orderings. From this information, are we able to recover the unique original orderings and matrix? We present an algorithm that identifies whether there is a unique ordering associated with a set of rows and columns, and outputs either the unique correct orderings for the rows and columns or the full collection of all valid orderings and valid matrices. We show that there is a constant $c > 0$ such that the algorithm terminates in $O(n^2)$ time with high probability and in expectation for random $n \times n$ binary matrices with i.i.d.\ Bernoulli $(p)$ entries $(m_{ij})_{ij=1}^n$ such that $\frac{c\log^2(n)}{n(\log\log(n))^2} \leq p \leq \frac{1}{2}$.
We study the problem of testing whether a symmetric $d \times d$ input matrix $A$ is symmetric positive semidefinite (PSD), or is $\epsilon$-far from the PSD cone, meaning that $\lambda_{\min}(A) \leq - \epsilon \|A\|_p$, where $\|A\|_p$ is the Schatten-$p$ norm of $A$. In applications one often needs to quickly tell if an input matrix is PSD, and a small distance from the PSD cone may be tolerable. We consider two well-studied query models for measuring efficiency, namely, the matrix-vector and vector-matrix-vector query models. We first consider one-sided testers, which are testers that correctly classify any PSD input, but may fail on a non-PSD input with a tiny failure probability. Up to logarithmic factors, in the matrix-vector query model we show a tight $\widetilde{\Theta}(1/\epsilon^{p/(2p+1)})$ bound, while in the vector-matrix-vector query model we show a tight $\widetilde{\Theta}(d^{1-1/p}/\epsilon)$ bound, for every $p \geq 1$. We also show a strong separation between one-sided and two-sided testers in the vector-matrix-vector model, where a two-sided tester can fail on both PSD and non-PSD inputs with a tiny failure probability. In particular, for the important case of the Frobenius norm, we show that any one-sided tester requires $\widetilde{\Omega}(\sqrt{d}/\epsilon)$ queries. However we introduce a bilinear sketch for two-sided testing from which we construct a Frobenius norm tester achieving the optimal $\widetilde{O}(1/\epsilon^2)$ queries. We also give a number of additional separations between adaptive and non-adaptive testers. Our techniques have implications beyond testing, providing new methods to approximate the spectrum of a matrix with Frobenius norm error using dimensionality reduction in a way that preserves the signs of eigenvalues.
We study the existence of finite characterisations for modal formulas. A finite characterisation of a modal formula $\varphi$ is a finite collection of positive and negative examples that distinguishes $\varphi$ from every other, non-equivalent modal formula, where an example is a finite pointed Kripke structure. This definition can be restricted to specific frame classes and to fragments of the modal language: a modal fragment $L$ admits finite characterisations with respect to a frame class $F$ if every formula $\varphi\in L$ has a finite characterisation with respect to $L$ consting of examples that are based on frames in $F$. Finite characterisations are useful for illustration, interactive specification, and debugging of formal specifications, and their existence is a precondition for exact learnability with membership queries. We show that the full modal language admits finite characterisations with respect to a frame class $F$ only when the modal logic of $F$ is locally tabular. We then study which modal fragments, freely generated by some set of connectives, admit finite characterisations. Our main result is that the positive modal language without the truth-constants $\top$ and $\bot$ admits finite characterisations w.r.t. the class of all frames. This result is essentially optimal: finite characterizability fails when the language is extended with the truth constant $\top$ or $\bot$ or with all but very limited forms of negation.
Given a conjunctive query $Q$ and a database $\mathbf{D}$, a direct access to the answers of $Q$ over $\mathbf{D}$ is the operation of returning, given an index $j$, the $j^{\mathsf{th}}$ answer for some order on its answers. While this problem is $\#\mathsf{P}$-hard in general with respect to combined complexity, many conjunctive queries have an underlying structure that allows for a direct access to their answers for some lexicographical ordering that takes polylogarithmic time in the size of the database after a polynomial time precomputation. Previous work has precisely characterised the tractable classes and given fine-grained lower bounds on the precomputation time needed depending on the structure of the query. In this paper, we generalise these tractability results to the case of signed conjunctive queries, that is, conjunctive queries that may contain negative atoms. Our technique is based on a class of circuits that can represent relational data. We first show that this class supports tractable direct access after a polynomial time preprocessing. We then give bounds on the size of the circuit needed to represent the answer set of signed conjunctive queries depending on their structure. Both results combined together allow us to prove the tractability of direct access for a large class of conjunctive queries. On the one hand, we recover the known tractable classes from the literature in the case of positive conjunctive queries. On the other hand, we generalise and unify known tractability results about negative conjunctive queries -- that is, queries having only negated atoms. In particular, we show that the class of $\beta$-acyclic negative conjunctive queries and the class of bounded nest set width negative conjunctive queries admit tractable direct access.
In the impartial selection problem, a subset of agents up to a fixed size $k$ among a group of $n$ is to be chosen based on votes cast by the agents themselves. A selection mechanism is impartial if no agent can influence its own chance of being selected by changing its vote. It is $\alpha$-optimal if, for every instance, the ratio between the votes received by the selected subset is at least a fraction of $\alpha$ of the votes received by the subset of size $k$ with the highest number of votes. We study deterministic impartial mechanisms in a more general setting with arbitrarily weighted votes and provide the first approximation guarantee, roughly $1/\lceil 2n/k\rceil$. When the number of agents to select is large enough compared to the total number of agents, this yields an improvement on the previously best known approximation ratio of $1/k$ for the unweighted setting. We further show that our mechanism can be adapted to the impartial assignment problem, in which multiple sets of up to $k$ agents are to be selected, with a loss in the approximation ratio of $1/2$.
A minimal perfect hash function (MPHF) maps a set of n keys to the first n integers without collisions. Representing this bijection needs at least $\log_2(e) \approx 1.443$ bits per key, and there is a wide range of practical implementations achieving about 2 bits per key. Minimal perfect hashing is a key ingredient in many compact data structures such as updatable retrieval data structures and approximate membership data structures. A simple implementation reaching the space lower bound is to sample random hash functions using brute-force, which needs about $e^n \approx 2.718^n$ tries in expectation. ShockHash recently reduced that to about $(e/2)^n \approx 1.359^n$ tries in expectation by sampling random graphs. With bipartite ShockHash, we now sample random bipartite graphs. In this paper, we describe the general algorithmic ideas of bipartite ShockHash and give an experimental evaluation. The key insight is that we can try all combinations of two hash functions, each mapping into one half of the output range. This reduces the number of sampled hash functions to only about $(\sqrt{e/2})^n \approx 1.166^n$ in expectation. In itself, this does not reduce the asymptotic running time much because all combinations still need to be tested. However, by filtering the candidates before combining them, we can reduce this to less than $1.175^n$ combinations in expectation. Our implementation of bipartite ShockHash is up to 3 orders of magnitude faster than original ShockHash. Inside the RecSplit framework, bipartite ShockHash-RS enables significantly larger base cases, leading to a construction that is, depending on the allotted space budget, up to 20 times faster. In our most extreme configuration, ShockHash-RS can build an MPHF for 10 million keys with 1.489 bits per key (within 3.3% of the lower bound) in about half an hour, pushing the limits of what is possible.
We introduce an active 3D reconstruction method which integrates visual perception, robot-object interaction, and 3D scanning to recover both the exterior and interior, i.e., unexposed, geometries of a target 3D object. Unlike other works in active vision which focus on optimizing camera viewpoints to better investigate the environment, the primary feature of our reconstruction is an analysis of the interactability of various parts of the target object and the ensuing part manipulation by a robot to enable scanning of occluded regions. As a result, an understanding of part articulations of the target object is obtained on top of complete geometry acquisition. Our method operates fully automatically by a Fetch robot with built-in RGBD sensors. It iterates between interaction analysis and interaction-driven reconstruction, scanning and reconstructing detected moveable parts one at a time, where both the articulated part detection and mesh reconstruction are carried out by neural networks. In the final step, all the remaining, non-articulated parts, including all the interior structures that had been exposed by prior part manipulations and subsequently scanned, are reconstructed to complete the acquisition. We demonstrate the performance of our method via qualitative and quantitative evaluation, ablation studies, comparisons to alternatives, as well as experiments in a real environment.
Bilingual Lexicon Induction (BLI) is a core task in multilingual NLP that still, to a large extent, relies on calculating cross-lingual word representations. Inspired by the global paradigm shift in NLP towards Large Language Models (LLMs), we examine the potential of the latest generation of LLMs for the development of bilingual lexicons. We ask the following research question: Is it possible to prompt and fine-tune multilingual LLMs (mLLMs) for BLI, and how does this approach compare against and complement current BLI approaches? To this end, we systematically study 1) zero-shot prompting for unsupervised BLI and 2) few-shot in-context prompting with a set of seed translation pairs, both without any LLM fine-tuning, as well as 3) standard BLI-oriented fine-tuning of smaller LLMs. We experiment with 18 open-source text-to-text mLLMs of different sizes (from 0.3B to 13B parameters) on two standard BLI benchmarks covering a range of typologically diverse languages. Our work is the first to demonstrate strong BLI capabilities of text-to-text mLLMs. The results reveal that few-shot prompting with in-context examples from nearest neighbours achieves the best performance, establishing new state-of-the-art BLI scores for many language pairs. We also conduct a series of in-depth analyses and ablation studies, providing more insights on BLI with (m)LLMs, also along with their limitations.
Fitting generative models to sequential data typically involves two recursive computations through time, one forward and one backward. The latter could be a computation of the loss gradient (as in backpropagation through time), or an inference algorithm (as in the RTS/Kalman smoother). The backward pass in particular is computationally expensive (since it is inherently serial and cannot exploit GPUs), and difficult to map onto biological processes. Work-arounds have been proposed; here we explore a very different one: requiring the generative model to learn the joint distribution over current and previous states, rather than merely the transition probabilities. We show on toy datasets that different architectures employing this principle can learn aspects of the data typically requiring the backward pass.
It is important to detect anomalous inputs when deploying machine learning systems. The use of larger and more complex inputs in deep learning magnifies the difficulty of distinguishing between anomalous and in-distribution examples. At the same time, diverse image and text data are available in enormous quantities. We propose leveraging these data to improve deep anomaly detection by training anomaly detectors against an auxiliary dataset of outliers, an approach we call Outlier Exposure (OE). This enables anomaly detectors to generalize and detect unseen anomalies. In extensive experiments on natural language processing and small- and large-scale vision tasks, we find that Outlier Exposure significantly improves detection performance. We also observe that cutting-edge generative models trained on CIFAR-10 may assign higher likelihoods to SVHN images than to CIFAR-10 images; we use OE to mitigate this issue. We also analyze the flexibility and robustness of Outlier Exposure, and identify characteristics of the auxiliary dataset that improve performance.
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