Four* papers on property testing last month! Lower bounds, upper bounds, distribution testing, quantum, and a new testing model!

* at least four. If we missed any, please let us know in the comments!

Lower Bounds for Convexity Testing, by Xi Chen, Anindya De, Shivam Nadimpalli, Rocco Servedio, and Erik Waingarten (arXiv). You’re given a membership oracle to a set \(S\) in \(\mathbb{R}^n\) (that is, query access to its indicator function \(f_S\colon \mathbb{R}^n\to \{0,1\}\)), and asked to decide if this set is convex, or “far from it”. This is a very natural and seemingly basic question— of course, we need to define what “far” means here, and the natural (normal, one may say) choice of underlying measure in \(\mathbb{R}^n\) is the standard Gaussian measure: \(d(S,T) = \Pr_{\mathcal{N}(0,I_n)}[ x \in S\Delta T]\).
Previous algorithms for this convexity testing question (and its tolerant testing analogue) are non-adaptive, and have \(2^{\tilde{O}(\sqrt{n})}\) query complexity. This paper shows that this is not just unfortunate, but also necessary: every non-adaptive tolerant tester for this question must make \(2^{\Omega(\sqrt[4]{n}}\) queries, and every (possibly adaptive) one-sided tester must have polynomial query complexity.

Replicable Uniformity Testing, by Sihan Liu and Christopher Ye (arXiv). In property testing, the algorithm must say YES with high probability on inputs which have the property, and NO with high probability on those which are far. On anything else, the algorithm is off the hook and can output either. This is typically considered to be fine, and, in any case, necessary to be able to obtain ultra-efficient algorithms. But what if, in this third case, we wanted to put the algorithm partially back on that hook, and required it to be consistent? The algorithm can answer either YES or NO, sure, but if I run it again on that same input, it should give the same answer with high probability. This is in line with a recent line of works on replicable algorithms, and is non-trivial to achieve in (the standard model of) distribution testing, where a distribution testing algorithm only gets to see random samples from the distribution, and thus needs to have a replicable behavior over that randomness. This paper introduces the question of replicable distribution testing, and provides both upper and lower bounds (essentially matching, with an asterisk) for the flagship task of uniformity testing.

Quantum property testing in sparse directed graphs, by Simon Apers, Frédéric Magniez, Sayantan Sen, and Dániel Szabó (arXiv). Graph property testing has a long and rich history in the classical setting, spanning more than two decades. There are several testing models, depending on whether the graph is dense, sparse, and directed or not: and even in the sparse, directed case, it is important to sometimes only allow outgoing edge queries. All these variants capture different meaningful scenarios, and relations and separations between them are known. This paper opens the direction of quantum testing for sparse graphs, either directed or not. The authors investigate what advantage quantum computers can bring for graph testing in this setting, and show one natural property for which a quadratic speedup exists: \(o(\sqrt{n})\) quantum queries in the outgoing-edge-query-only (unidrectional) sparse model, while classically \(\Omega(n)\) are necessary. They also show that this is not always the case: quantum testing of 3-colorability, as in the classical case, does not admit a \(o(n)\)-query tester.

Relative-error monotonicity testing, by Xi Chen, Anindya De, Yizhi Huang, Yuhao Li, Shivam Nadimpalli, Rocco Servedio, and Tianqi Yang (arXiv). Property testing of Boolean functions is defined “absolutely“: the distance between two functions is the fraction of the domain on which they differ, i.e., \(\displaystyle\frac{|f^{-1}(\{1\})\Delta g^{-1}(\{1\})|}{2^n}\)
This makes sense when the functions have a reasonable number of satisfying assignments: but may be much less meaningful for sparse functions, which only are non-zero on a \(o(1)\) fraction of the inputs—for instance, functions where “all the action” is concentrated in a tiny subcube of the hypercube. All these functions are vanishingly close to each other! To address this, the authors introduce a new distance notion, relative-error, where the distance from \(g\) to \(f\) is scaled by the sparsity of \(f\):
\(\displaystyle\frac{|f^{-1}(\{1\})\Delta g^{-1}(\{1\})|}{|f^{-1}(\{1\})|}\)
This requires a slightly different access model to avoid trivial impossibility results, so the tester is augmented with sampling access to satisfying assignments of \(f\), on top of query access to \(f\) (as otherwise it may just never even find one satisfying assignment). After introducing and motivating this testing model, the paper initiates its study in the specific case of testing monotonicity of Boolean functions.

So, we had a pretty fantastic September. Besides the fact that September saw eight papers, the Computer Science community also bagged two Nobel Prizes(!) — reactions to which are kind of mixed from what I see around me. Anyhow without further delay, let us circle back to property testing. So, eight papers, yes. Without further ado, let us look at our spread.

Public Coin Interactive Proofs for Label-Invariant Distribution Properties by Tal Herman (ECCC) Suppose you have an unknown distribution \(\mathcal{D}\) supported over \([N]\). Suppose I claim that this distribution has entropy \(\texttt{blah}\). You have sample access to \(\mathcal{D}\) and you want to check my claim. To this end, you decide to engage me, a suspicious shady prover, in an Interactive Protocol where you, the verifier is restricted to use public coin tosses. The main result of this paper asserts that you can do this with a mere \(\widetilde{O}(N^{2/3})\) samples. You also get the same bound on the communication complexity. What’s more is that you get algorithms with running time of the same order. What’s even more, is that you get similar algorithms for all label invariant properties of \(\mathcal{D}\). You can contrast this with the seminal result of Valiant and Valiant from 2011 which asserts that you can approximate the distance between your input distribution \(\mathcal{D}\) and the label invariant property of interest (without any prover) using \(\Theta(N/\log N)\) samples. So, this result shows that the computation under the interactive model is more efficient than standalone computation even in the public coin toss model.

How to Verify Any (Reasonable) Distribution Property: Computationally Sound Argument Systems for Distributions by Tal Herman and Guy Rothblum (arXiv) Let us consider the same setup as above. Again, you have an unknown distribution (supported over \([N]\)) which you have sample access to and I assert that this distribution has a certain property. You want to verify whether my claim is correct or bogus. The main algorithmic result of this paper has a cryptographic ring to it: Assuming the existence of collision resistant hash functions, the authors show that for any reasonable distribution property, you have set up an interactive protocol where the verifier can decide whether or not the prover’s claim is bogus using \(\widetilde{O}(\sqrt N)\) samples. Also, the communication complexity is of the same order.

Random local access for sampling k-SAT solutions by Dingding Dong and Nitya Mani (arXiv) I find this paper pretty intriguing. So, here is the setup. I give you some \(k\)-SAT instance and I promise that no variable in this instance shows up in more than \(d\) clauses. Recall that for \(d \leq 2^k/(2e k)\), Lovasz Local Lemma assures you that there exists a satisfying assignment. What’s more is that the famous Moser-Tardos algorithm even allows you to find one satisfying assignment in polynomial time. In a beautiful work, in the regime where \(d \leq 2^{ck}\) (where \(0 < c < 1\) is a sufficiently small constant), Moitra gave samplers which return an almost uniformly random satisfying assignment. Note that this is not possible direct adaptation the Moser-Tardos algorithm. Anyhow, back to the setting considered in this work. So, consider the following sublinear challenge. Denote by \(\mu\) the uniform distribution supported over all satisfying assignments to the given \(k\)-SAT instance where each variable shows up in more than \(d \leq 2^{ck}\) clauses. I want you to sample the assignment \(\sigma(v)\) for a variable \(v\) from the marginal \(\mu_v\) induced by \(\mu\) on \(v\) fast (in \(poly(\log n)\) time) in expectation. Of course this has to be done in some consistent fashion overall which is very LCAish in flavor and I will not detail that here. Rest assured the featured paper rises up to the challenge.

New Direct Sum Tests by Alek Westover, Edward Yu, Kai Zheng (arXiv) We say that a \(\mathbb{F}_2\)-valued function \(f\) over the \(d\)-dimensional grid \(f \colon [n]^d \to \mathbb{F}_2\) is a direct sum if there are \(d\) one-dimensional functions \(L_i \colon [n] \to \mathbb{F}_2\) such that \(f(x) = \sum_{i=1}^d L_i(x_i)\). In a paper we covered in October 2019, Dinur and Golubev presented an algorithm for direct sum testing and left its analysis for future research. The featured paper analyzes this test and shows that it is indeed a bonafide direct sum tester. I omit the description of the tester. I will just leave with one note — this looks like an appetizing read for Boolean Fourier Analysis aficionados.

Directed Hypercube Routing, a Generalized Lehman-Ron Theorem, and Monotonicity Testing by Deeparnab Chakrabarty and C. Seshadhri (arXiv) This is a very refreshing read with (our very own) Seshadhri as one of the authors. Chakrabarty and Seshadhri have teamed up (often in a trio with Hadley Black) in their attempts to understand the deep dark secrets of testing boolean monotonicity over the boolean hypercube. The featured paper conjectures a bold generalization of Lehman-Ron theorem over the hypercube and suggests that the conjectured routing once established could pave the path forward towards a more transparent understanding of directed isoperimetric inequalities over the boolean hypercube. In my opinion, it is worth your while to check this one out.

Lempel-Ziv (LZ77) Factorization in Sublinear Time by Dominik Kempa and Tomasz Kociumaka (arXiv) A pretty unique topic. The main result of this paper is what you see in the title — after 50 years since its introduction, finally we can factorize LZ77 in sublinear time. If you are like me, you might be asking but what does that mean? Quoting from the abstract

Lempel-Ziv (LZ77) factorization is a fundamental problem in string processing: Greedily partition a given string T from left to right into blocks (called phrases) so that each phrase is either the leftmost occurrence of a letter or the longest prefix of the unprocessed suffix that has another occurrence earlier in T.

Indeed, the abstract has more fascinating information about this problem. Quoting again.

Sublinear-time algorithms are known for nearly all other fundamental problems on
strings, but LZ77 seems resistant to all currently known techniques.

Looks like new year came early for PTReview readers. This paper promises to be fun by the contents I sampled so far.

Computing String Covers in Sublinear Time by Jakub Radoszewski and Wiktor Zuba (arXiv) A substring (actually prefix) \(C\) is called a cover of text \(T\) is \(T\) can be constructed by concatenations and superpositions of \(C\). Suppose the text string \(T\) contains \(n\) symbols from some alphabet of fixed size. The main theorem of the featured paper asserts that given a string \(T\), you can find a representation of all of its covers (at most \(n\) of them — they are all prefixes) in merely \(O(n/\log_{\sigma} n)\) time. This bound is also shown to be optimal — indeed the authors show that you cannot compute a representation (in a certain model formulated by Charalampopoulos et al in FOCS 2020) for the shortest cover in less than \(o(n/log n)\) time.

Quantum Channel Testing in Average-Case Distance by Gregory Rosenthal, Hugo Aaronson, Sathyawageeswar Subramanian, Animesh Datta and Tom Gur (arXiv) This paper considers a new diversion in quantum land. Namely, it considers the task of testing quantum channels. The setup is this: I have a \(d\)-dimensional quantum system which is just a \(\mathbb{C}^{d \times d}\) psd matrix over complex numbers with trace \(1\). A quantum channel is a just a linear transformation that maps \(d_{in}\) dimensional quantum systems to a \(d_{out}\)-dimensional quantum system. With repsect to some appropriate norm (the so-called diamond norm), one of the results in this paper proves a \(poly(d_{in})/\varepsilon\) lower bound for testing identity to any fixed channel in diamond distance. This lower bound is shown to hold in a very strong query model appropriate for the quantum setting.

This month we have only one paper. I imagine the community will be on fire when we come back with our news for August.

Constant Degree Direct Product Testers with Small Soundness by Mitali Bafna, Noam Lifshitz, and Dor Minzer (arXiv) (Please excuse my lack of technical comfort with the contents of this paper. Corrections Welcome) As the title indicates, and the authors also emphasize, the primary goal of the featured paper is to construct direct product testers with constant degree. Let us try to unpack this a little by first understanding what does a direct product test mean. So I have this function \(f \colon [n] \to \{0,1\}\). I give you access to this function in an indirect way via a table \(F\) to which you have query access. The central task in direct product testing is to check whether \(F\) is a valid encoding of \(f\) by querying \(F\) on a small number of locations. This paper focuses on those properties which you can test with two queries. Dinur and Kaufman noted that high dimensional expanders can be leveraged towards obtaining \(2\)-query direct product testers. The main result of this paper shows that there are high dimensional expanders for which the Dinur-Kaufman direct product test has small soundness.

A nice diverse month, this June 2024. We have a selection of papers on quantum property testing, shortest paths, polynomial threshold functions, and nearest neighbor search. And a nice survey on a closely related topic. Onward with the papers!

Invitation to Local Algorithms by Václav Rozhoň (arXiv). This is a great survey on local algorithms, a topic (roughly) about message passing distributed algorithms on graphs. While that is not a sublinear algorithm per se, the core combinatorial question has relevance to our readers. Informally, a “local problem” is one where, if a solution is incorrect, that can be determined by looking at a small radius around some vertex. For example, consider problems like maximal independent set, matching, etc. The problem is to construct such a solution in a distributed manner by only looking at low radius balls. There is a direct connection with Local Computation Algorithms (LCAs), with the primary difference being the complexity resource. In Local Algorithms as defined, one is interested in the number of rounds of computation, corresponding to the radius of balls. In LCAs, we care about the number of vertices seen, which is the size of the balls. The survey gives a detailed discussion of these connections.

Testably Learning Polynomial Threshold Functions by Lucas Slot, Stefan Tiegel, and Manuel Wiedmer (arXiv). Consider the classic model of agnostic learning with a distributional assumption. Our aim is to learn a function \(f: X \to \{-1,1\}\). The learner gets labeled samples \((x,f(x))\), where \(x\) is drawn from an unknown distribution \(\mathcal{D}\). In agnostic learning, we aim to output a hypothesis \(\hat{f}\) such that \(\|f – \hat{f}\|\) is small, where distance is measured according to \(\mathcal{D}\). Often one makes distributional assumptions (like \(\mathcal{D}\) is Gaussian) to get non-trivial results. This is a severe limitation, since such distributional assumptions cannot be validated. A recent model of Rubinfeld-Vasilyan asks for “tester-learner” pairs that address this issue. We first run a “tester” to check the property (say, Gaussian) of the distribution. If the tester passes, then we run the learner. If \(\mathcal{D}\) satisfies the distributional property, the tester is guaranteed to pass (whp). If the tester passes, then the learner is guaranteed to output a valid hypothesis. Thus, the tester is allowed to pass distributions that may be far from satisfying the property, but in such situations, the learner outputs a correct hypothesis. This paper gives such tester-learner pairs for learning polynomial threshold functions with polynomially many samples. Previously, such learning results were not known even for degree-2 PTFs with respect to the Gaussian distribution.

A Sublinear Algorithm for Approximate Shortest Paths in Large Networks by Sabyasachi Basu, Nadia Kōshima, Talya Eden, Omri Ben-Eliezer, C. Seshadhri (arXiv). This paper brings a mix of a classic problem, a real-world observation, and a new graph class. Finding shortest paths in graphs is about as classic as it gets. There is a plethora of applied work on shortest paths algorithms for real-world networks, much of which is based on landmarking. One precomputes distances between chosen landmarks, and then uses those to output shortest path distance queries. This paper asks: is it possible to get approximation shortest paths in sublinear time? Hence, we are not even allowed to preprocess the entire graph. This paper builds on previous work in social networks about “core-periphery” structure. The core is a denser, small region of the graph; the rest of the vertices form the periphery that connect to the core. The hypothesis is that shortest paths in real-world graphs typically go via the core. Given the (small) core, one can build small sublinear data structures that quickly answer shortest path queries. The paper uses insights from previous work on core-periphery structure, and gives a practical sublinear algorithm for computing approximate shortest paths. If the graph comes from a Chung-Lu distribution, then queries can be provably answered in \(n^{o(1)}\) time.

Quantum Property Testing Algorithm for the Concatenation of Two Palindromes Language by Kamil Khadiev and Danil Serov (arXiv). An early result of property testing is that all regular languages can tested in constant time. But the simple (non-regular) language \(\mathcal{L}\) of strings formed by concatenating two palindromes requires super-constant queries. Specifically, the complexity of property testing \(\mathcal{L}\) is \(\Theta(\sqrt{n})\) (ignoring log and \(\varepsilon\) factors). Here, \(n\) denotes the string size. This paper shows that there exist quantum property testing algorithms that beat this bound. There is a quantum property tester for the language \(\mathcal{L}\) that makes \(\widetilde{O}(n^{1/3})\) queries. The paper also shows that quantum query complexity of the exact decision problem is \(\Theta(\sqrt{n})\). Thus, one gets non-trivial quantum speedup for both the property testing and exact problem.

A Bi-metric Framework for Fast Similarity Search by Haike Xu, Sandeep Silwal, and Piotr Indyk (arXiv). Nearest-neighbor (NN) search is one of the most important algorithmic tasks in modern data analysis. The usual setting is that we have a set of \(n\) points over a metric, denoted by \(D\). We preprocess the points and construct a data structure. Given a query point \(q\), we wish to find the (approximate) closest points from our data set, in time sublinear in the data size. This paper introduces a bi-metric setting. Think of \(D\) as expensive to compute, while there exists a cheap “proxy metric” \(d\). For example, if the points represent images, the expensive metric \(D\) could be some complex (but semantically accurate) metric, while \(d\) may represent a simple Euclidean metric over an embedding. Formally, \(d\) is a (large) constant factor approximation of \(D\). This paper gives an algorithm that performs nearest neighbor search, but only preprocesses over \(d\). Given, a query \(q\), it makes sublinear queries to the expensive \(D\). All in all, it is a truly sublinear algorithm in terms of \(D\), and leverages the preprocessing over the cheap metric \(d\). Technically, this paper gives a meta-algorithm that can be applied to any “base” NN algorithm. The paper proves results for two popular NN algorithms, and gives extensive empirical evidence for the underlying approach.

May came with 3 new papers on property testing algorithms — or inspired by them.

Interactive Proofs for General Distribution Properties, by Tal Herman and Guy Rothblum (ECCC). Following a fruitful line of work (including by the authors themselves: see, e.g., this previous monthly post), this paper considers interactive proofs for distribution testing: Merlin and Arthur have data over a universe of size \(n\), Arthur wants to test properties of that data (probability distribution), but he has much less data (samples) than Merlin.
As it turns out, as long as the property he’s interested in can be checked efficiently (computationally: via a small-depth circuit), then Arthur can do it with strongly sublinear sample complexity: he needs only \(n^{1-\Omega(1)}\) samples, even for tolerant testing! And all that’s needed is a small number of rounds of interaction with Merlin. And even more, all (honest) parties can do that via a computationally efficient protocol…

Oracle-Checker Scheme for Evaluating a Generative Large Language Model, by Yueling Jenny Zeng, Li-C. Wang, and Thomas Ibbetson (arXiv). This paper draws inspiration from property testing and program checking (à la Blum, Luby, and Rubinfeld) to check the output of large language models (LLMs): specifically, for the task of entity extraction: the authors formalize how to view entity extraction as a homomorphism, and then assess empirically what using a property tester for linearity leads to. Overall, it sounds like an interesting (and somewhat unexpected?) use of property testing for LLM trustworthiness assessment!

Property testing in graphical models: testing small separation numbers, by Luc Devroye, Gábor Lugosi, and Piotr Zwiernik (arXiv). Here too, ideas from property testing are used, this time in the context of high-dimensional (Gaussian) graphical models. This paper focuses on testing properties of the structure of the graphical model: given query access to the covariance matrix \(\Sigma\) consistent with some underlying graph structure \(G\), can we test whether this structure is a tree? Is it has small separation number?
The focus differs a little from the classical setting of property testing, in that there is no distance parameter and the goal is to get an exact decision algorithm (adaptive, but with unbounded query complexity: rejecting graphs that are far from the property as a function of the unknown distance parameter, and always accepting graphs with the property). But besides this small variation, great to see more uses of property testing in the wild!

We have seven papers for you this month. Our potpourri includes two papers apiece on each of the following themes: Distribution Testing, property testing with a quantum twist, graph property testing, and finally two papers on testing function properties. A featured paper this month covers progress on optimal non-adaptive junta testers. Without further ado, let’s dig in. As usual, please let us know if I missed any papers.

Testing \(C_k\) freeness in bounded-arboricity graphs by Talya Eden, Reut Levi, and Dana Ron (arXiv) Our post from July 2021 highlighted an open problem posed by Goldreich. This problem asks if it is possible to transform property testers for bounded degree graphs to property testers for unbounded degree graphs with general arboricity. The featured paper answers the question Goldreich posed in the negative. In particular, testing \(C_4\) freeness in bounded arboricity graphs (with possibly unbounded degree) already admits an \(\Omega(n^{1/4})\) one-sided lowerbound. Up to \(\log\) factors, the paper also proves a matching upperbound. The same bounds hold for \(C_5\)-freeness. Further, for every \(k \geq 6\), the paper proves an \(\Omega(n^{1/3})\) one-sided lower-bound.

Directed Isoperimetry and Monotonicity Testing: A Dynamical Approach by Renato Ferreira Pinto Jr (arXiv) The featured paper considers a classic of property testing. So, you want to test whether an input real-valued function \(f \colon [0,1]^d \to \mathbb{R}\) over the solid cube is monotone or whether it is \(\varepsilon\)-far from being monotone. It is convenient to require that the input function better be differentiable and Lipschictz (for those of you keeping scores, this should remind you of our post from July 2023 where we covered a previous work in the same setting again by Renato). Motivated by the success of testers developed for the boolean hypercube domain, a natural analogy suggests to have in addition to a standard value oracle, an additional derivative oracle which takes as input a point \(x \in [0,1]^d\) and a direction \(\mathbf{v} \in \mathbb{R}^d\). This oracle allows you to query directional derivative at \(x\) along \(\mathbf{v}\). Renato’s punchline is a directed Poincare Inequality which in the above setup, connects the distance to monotonicity to the square root of directed analog of a suitable notion of influence. The techniques used in the paper seem intriguing. They are inspired by the original proof of the classic Poincare Inequality. In particular, as Renato notes “the main theme of our proof of this result is the study of the convergence of a partial differential equation.”

Optimal Non-Adaptive Tolerant Junta Testing via Local Estimators by Shivam Nadimpalli, Shyamal Patel (arXiv) The problem of Tolerant Junta testing is no stranger to our regular readers. This paper is fairly intriguing for those of you who care about testing function properties. As the abstract notes, this is the first paper to nail down the true (non-adaptive) query complexity for tolerantly testing some natural property of boolean functions. The main result of this paper presents a lower bound of \(2^{\widetilde{\Omega}(\sqrt{k \log(1/\varepsilon)})}\) on the task of non-adaptive tolerant \(k\)-junta testing. The paper presents an almost matching non-adaptive algorithm as well.

Testing Intersectingness of Uniform Families by Ishay Haviv and Michal Parnas (arXiv) Let \(\mathcal{F}\) denote an intersecting family all sets in which are subsets of an underlying \(n\)-element universe. This means that for any \(F_1, F_2 \in \mathcal{F}\), you have that \(F_1 \cap F_2 \neq \emptyset\). Some of you might immediately recall Erdos-Ko-Rado theorem which asserts an upperbound on the size of such an intersecting family where every set has size \(k\). Another famous result is Lovasz’s (positive) resolution of Kneser’s conjecture which asserts an lowerbound on the number of intersecting families you need to cover all \(k\)-subsets of \([n]\). For ease of discussion, let us follow the authors and cook up a property testing problem \(\textsf{INTERSECTING}_{n,k, \varepsilon}\). In this problem, you are given access to the indicator \(f \colon {[n] \choose k} \to \{0,1\}\) encoding the family \(\mathcal{F}\) and you ask whether \(\mathcal{F}\) is intersecting or whether it is \(\varepsilon\)-far from it. Recently, Chen-De-Li-Nadimpalli-Servedio explored the non-uniform-set-size variant of this problem (which we covered here). They presented one-sided algorithms with a non-adaptive query bound of \(2^{\widetilde{O}(\sqrt{n \log(1/\varepsilon)})}\) for this problem and they also showed an almost matching lowerbound. The featured paper contrasts these results with the situation you obtain when all the set sizes are indeed the same (that is, the paper explores the testability of \(\textsf{INTERSECTING}_{n,k, \varepsilon}\). Of the numerous results in the paper, let me highlight one: you can test \(\textsf{INTERSECTING}_{n,k, \varepsilon}\) with two-sided error with a mere \(O\big(\log n \cdot \frac{1}{\varepsilon} \big)\) queries. This result holds for \(\varepsilon \geq \Omega(k/n)^r\) for a fixed \(r\).

Distribution-Free Testing of Decision Lists with a Sublinear Number of Queries by Xi Chen, Yumou Fei and Shyamal Patel (arXiv) Decision lists are a popular and convenient way to represent a boolean function which is learnable from examples. In more detail, a decision list is a collection of pairs \((\alpha_1, \beta_1), (\alpha_2, \beta_2), \ldots (\alpha_k, \beta_k)\) (here, \(\alpha_j\)’s denote literals and \(\beta_j\)’s are bits). This list defines a boolean function \(f \colon \{0,1\}^n \to \{0,1\}\) as follows: for any \(\mathbf{x} \in \{0,1\}^n\), you let \(f(x) = \beta_j\) where \(j\) is the smallest index such that \(\alpha_j\) is satisfied by \(\mathbf{x}\). With this (not super rigorous) definition out of the way, I can now tell you about the main result of this featured paper. The main theorem of this paper asserts that in the distribution-free framework for property testing, you still get sublinear time algorithms for testing decision lists. In particular, thanks to this paper, now you can engineer a two-sided adaptive distribution free algorithm for testing decision lists which makes runs in time \(\widetilde{O}(n^{11/12})\).

Simple algorithms to test and learn local Hamiltonians by Francisco Escudero Gutiérrez (arXiv) The featured paper considers the task of (tolerantly) testing and learning an \(n\)-qubit \(k\)-local Hamiltonian from queries to its evolution operator. The main result of the paper asserts that the task of Tolerant Hamiltonian Locality Testing can be done with a mere \(1/(\varepsilon_2 – \varepsilon_1)^8\) queries to the evolution operator.

Local Test for Unitarily Invariant Properties of Bipartite Quantum States by Kean Chen, Qisheng Wang, Zhicheng Zhang (arXiv) Lots of investigations on quantum entanglement consider bipartite quantum states. The featured paper considers the task of testing properties of bipartite pure states. The paper begins by recalling a helpful duality between a property of bipartite pure states being unitarily invariant on one part, and the property being locally testable on the other part. This duality does not offer any insights into the query complexity of the local tester. The main result of the paper proves that the local tester indeed achieves optimal sample complexity over all global testers.

Learning Intersections of Halfspaces with Distribution Shift: Improved Algorithms and SQ Lower Bounds by Adam R. Klivans, Konstantinos Stavropoulos, Arsen Vasilyan (arXiv) Consider the standard setup of supervised learning with a twist. Namely, the distribution from which you receive samples in the training phase (\(\mathcal{D}\)) and the distribution from which the samples are taken in the test phase (\(\mathcal{D}’\)) are not necessarily the same (and hence the name — distribution shift). For example, this situation may arise if you use one set of equipments in the training phase and another during the test phase. This model was explored in a previous work by the same set of authors where they considered the task of obtaining a low-error classifier (on \(\mathcal{D}’\)) when they are additionally told that the training samples pass some helpful test. In this paper, the authors explore the problem of testing intersections of halfspaces with respect to Gaussian Training Distributions. The main contribution of the paper is a set of new positive results which extend the results from PAC learnability to the learnability under the new model. Indeed, under some reasonably mild assumptions, the bounds in the new model match the bounds from the standard PAC model. For quantitative details, please refer to the paper.

We have a rich bounty of papers this March (with a Feb 29 submission that got associated with this month). Local Computations Algorithms, Error Correction, PageRank computations, Quantum testing, Distribution testing, Graph property testing, and nice note on a statement we all knew (but never bothered to prove!). (Ed: We missed a paper on graph property testing, and updated the blurb on “A basic lower bound for property testing”.)

Average-Case Local Computation Algorithms by Amartya Shankha Biswas, Ruidi Cao, Edward Pyne, and Ronitt Rubinfeld (arXiv). Readers of this blog are likely familiar with Local Computation Algorithms (LCA) model. Imagine the input is (say) a large undirected graph \(G\), to which an algorithm has query access to. The aim is to give sublinear access to a large, linear sized output, such as a graph spanner. So each bit/coordinate/edge of the output should be computable with only sublinear access to \(G\) (these are called probes). There is a rich line of results for computing \(k\)-spanners, which are sparse subgraphs of \(G\) that approximate the shortest path distances up to a factor \(k\). Previous LCA lower bounds give a barrier of \(\Omega(\sqrt{n})\) probes per spanner query. To beat this bound, this paper introduces average-case LCAs. Imagine that \(G\) is generated according to graph distribution, like Erdős-Rényi, and the LCA only needs to succeed with high probability over the input. For such settings, this paper gives a number of LCAs for different parameter settings that beats the \(\sqrt{n}\) probe barrier. An additional concept introduced is that of joint sampling. Here, the LCA is expected to generate the random input \(G\) and gives access to the spanner. It is shown that one can beat the trivial bound obtained by just coupling together LCAs for input generation and spanner construction.

Local Correction of Linear Functions over the Boolean Cube by Prashanth Amireddy, Amik Raj Behera, Manaswi Paraashar, Srikanth Srinivasan, and Madhu Sudan (ECCC, arXiv). Speaking of LCAs, a classic example of an LCA is a local error correcting code (LCC). The input is a function \(f:\{0,1\}^n \to G\) where \(G\) is an Abelian group. Think of codewords as linear functions over this space. A \((\delta, q)\)-LCC would provide local access to the closest keyword to \(f\) with distance \(< \delta\) from the property of linear functions. Each coordinate of the closest keyword is obtained with \(q\) queries/probes to \(f\). Classic results provide a \((\Omega(1), O(1))\)-LCC when \(G = \mathbb{Z}_2\). The aim of this paper is to consider general range groups, such as the reals. Not much is known in this space for LCCs. This paper gives a \((1/4-\varepsilon, \widetilde{O}(\log n))\)-LCC for this setting. (Note that \(1/4\) is the decoding radius.) Most previous LCC results use “common” symmetries among the (hypercube) domain and range, while this result works when the range is totally unrelated to the domain. Moreover the \(\widetilde{O}(\log n)\) query complexity is nearly matches the latex \(\Omega(\log n)\)-query lower bound for this problem.

Revisiting Local Computation of PageRank: Simple and Optimal by Hanzhi Wang, Zhewei Wei, Ji-Rong Wen, and Mingji Yang (arXiv). PageRank is one of the fundamental computations in modern network science. Given a (possibly) directed graph \(G = (V,E)\), PageRank defines a “truncated random walk” that works as follows. It starts from the uniform distribution. At each step, independently with probability \(\alpha\), the walk just stops. With the remaining probability, it performs a single random walk step by going to a random (out) neighbor. (This definition is equivalent to the stationary distribution of the random walk Markov Chain with a teleport probability \(\alpha\).) The PageRank value of vertex \(t\), denoted \(\pi(t)\), is the probability of ending at \(t\). The obvious algorithm for estimating \(t\) is to simply perform a bunch of random walks, requiring \(\Theta(1/\pi(t))\) to get non-trivial estimates. The average PageRank value is \(1/n\), so this leads to a linear time algorithm for an average vertex. This is a rich history of work on beating this bound, and getting a sublinear query complexity for (almost) all vertices. The approach is to get an estimator with running time (ideally) \(O(n \pi(t))\), By combining both estimators, we get \(O(\sqrt{n})\) time algorithms for estimating PageRank. This paper gets as close as possible to this bound, and achieves (the almost optimal) bound of \(O(\sqrt{n} \cdot m^{1/4})\). The actual bound is more involved and depends on the maximum in and out degrees. It is worth noting that random walks on directed graphs are nasty objects, so getting these algorithms is extremely challenging. Moreover, this paper nails down the upper and lower bounds in terms of the \(n, m\), and the max degrees.

Efficient Algorithms for Personalized PageRank Computation: A Survey by Mingji Yang, Hanzhi Wang, Zhewei Wei, Sibo Wang, and Ji-Rong Wen (arXiv). This survey isn’t meant for a TCS audience per se, but has bearing to the previous paper. And it has many explicit mentions to sublinear algorithms for PageRank computations. This survey focuses more on Personalized PageRank (PPR), wherein the walks starts from a single source vertex. Section 7 talks about sublinear algorithms for estimating individual PPR values, and discusses the techniques involved in these algorithms. This is a good survey for getting a sense of the interesting problems in estimating (P)PR values.

New Graph and Hypergraph Container Lemmas with Applications in Property Testing by Eric Blais and Cameron Seth (arXiv). The container method is a combinatorial approach that is now seeing many property testing applications. The best way to motivate this is to consider the classic problem of property testing if a dense graph \(G\) has a clique of size \(\rho n\). The canonical tester samples a small set of vertices (of size \(f(\rho)/poly(\varepsilon)\), where \(f(\cdot)\) is some explicit function), computes the largest clique in this set, and then extrapolates that to guess the largest clique size in \(G\). If \(G\) has a clique of size at least \(\rho n\) (the YES case), the analysis is an easy exercise. If \(G\) is \(\varepsilon\)-far from having a large clique (the NO case), the analysis needs to deal with probability that small cliques in \(G\) lead to large cliques in the sample (just by random deviation). This argument needs a union bound over “all possible (small) cliques” of \(G\). And this is always the hard part. The container method is about proving that there is a small (polynomial) number of “containers”, that contain all small cliques. The union bound then works out with stronger parameters. This was introduced to property testing in a beautiful, previous work of the authors, which led to substantial improvements for many property testing problems over dense graphs. In this paper, they generalize the container method to more settings, and prove stronger property testing bounds for satisfiability and \(k\)-colorability.

Hamiltonian Property Testing by Andreas Bluhm, Matthias C. Caro, and Aadil Oufkir (arXiv). This paper is on property testing the quantum notion of Hamiltonian locality. This description will only highlight my poor knowledge of quantum physics, but let me try. A Hamiltonian is an operator on \(n\) qubits, each of which should be thought of as a complex number with magnitude \(1\) (you can think of this a 2D vector, with each axis representing a classical bit). On a single qubit, there are four possible operations: identity, negate the complex part, switch the real and complex parts, or negate the complex and switch real/complex parts. These operations can be represented as \(2 \times 2\) matrices (the qubit is a 2D vector), and form a basis. We can generalize this basis to \(n\) qubits as follows. Pick a string in \(\{0,1,2,3\}^n\), where \(0\) indicates the identity operation, and the others map to the three non-trivial qubit transforms above. Apply the corresponding operation to the corresponding qubit. Mathematically, we represent this operation as a tensor product of the corresponding \(2 \times 2\) matrices. These operators form the Pauli basis; the locality of a basis operator is the number of non-trivial qubit operators used. Any Hamiltonian can be written in the Pauli basis, and it is called \(k\)-local if it is spanned by at most \(k\)-local basis operators. This paper sets up a natural property testing question. Distinguish the input Hamilton being \(k\)-local from the input being \(\varepsilon\)-far from being \(k\)-local. Access to the Hamiltonian (and exponentially large object to “write down”) is achieved by applying the operator/transformation to a vector of qubits. If distance between operators is measured by \(l_\infty\)-norm, then the property testing problem requires exponentially many queries to the Hamiltonian. The main result is that if distance is measure by \(l_2\)-norm, then only polynomially many queries suffice to test \(k\)-locality, for constant \(k\).

Equivalence Testing: The Power of Bounded Adaptivity by Diptarka Chakraborty, Sourav Chakraborty, Gunjan Kumar, Kuldeep S. Meel (arXiv). We all know the problem of equivalence/identity testing of distributions. This work is on the conditional sampling model. Consider two input distributions \(\mathcal{P}\) and \(\mathcal{Q}\) over domain \([n]\). For any subset \(S\) of the domain, we can generate a sample from the input distributions, conditioned on being in \(S\). In this model, existing results show that equivalence (\(\mathcal{P} = \mathcal{Q}\)) can property tested using \(O(\log\log n)\) samples (ignoring \(\varepsilon\) dependencies), a significant demonstration of the power of conditional sampling. But this is a sequential, and hence adaptive, algorithm. The best known bound for non-adaptive adaptive algorithms is \(O(\log^{12}n)\), while the known lower bound is \(\Omega(\log n)\). This paper shows that with a single round of adaptivity, one can get a \(\widetilde{O}(\log n)\)-query equivalence tester. This result opens up a rich source of problems for the conditional sampling model, wherein we look at the power of bounded adaptivity.

On the query complexity of testing local graph properties in the bounded-degree graph model by Oded Goldreich (arXiv). Consider property testing in bounded degree graph model. The input graph \(G = ([n], E)\) is represented as an adjacency list. In the dense graph setting, a vast majority of “interesting” properties are testable in time independent to graph size. For bounded-degree graphs, the story is much more complicated. A fundamental question is to characterize such “constant-time” testable properties. A local property (in this context) is one that is basically defined as subgraph freeness, with some augmentations by vertex degree. Previous work suggested that all local properties can be tested in time independent on graph size. This conjecture was subsequently refuted, but with a non-constructive argument that does not give an explicit query lower bound. This paper overcomes the non-constructive barrier, and gives a query lower bound of \(\Omega(\log^{(4)} n)\) for an explicit local property (\(\log^{(i)}\) is the iterated log). In a slightly weaker model, where the algorithm is not given the exact size of \(G\), the lower bound can be significantly improved to \(\Omega(\log n)\).

A basic lower bound for property testing by Eldar Fischer (arXiv). We end with a fundamental statement that you must have known. In any (discrete) setting, consider some non-trivial property \(\mathcal{P}\) with distance measured by Hamming distance. The setting that covers almost all property testing settings. Of course, any property tester requires \(\Omega(1/\varepsilon)\) queries. Intuitively, if a tester makes less \(O(1/\varepsilon)\) input queries, it might not “touch” the portion of the input that helps it in the distinguishing task. This basic fact has not been proven explicitly, and this note resolves that issue. The short paper gives a full self-contained proof of this fact, and does so without resorting to Yao’s minimax lemma. The proof is not difficult, but needs some care because the tester could be adaptive. Great read for students. (Further addendum: Nader Bshouty and Oded Goldreich pointed out that a previous note by Bshouty and Goldreich also proves such a lower bound. These results differ on the precise notation of a “non-trivial property”. Fischer’s definition is more direct and slightly stronger than the Bshouty-Goldreich definition. The Bshouty-Goldreich proof uses a construction of a “sufficiently separated” set of strings, which is different from the proof in this note. Again, both proofs are excellent reads for students to learn more about property testing. These results give a good perspective on crafting the right definitions.)

February this year was slightly superlinear, with 29 days instead of the usual 28. As a result… 5 property testing papers! (Including one overlooked from January),

Testing Calibration in Subquadratic Time, by Lunjia Hu, Kevin Tian, and Chutong Yang (arXiv). The authors consider the question of model calibration, where a binary predictor is said to be calibrated if \(\mathbb{E}[ y\mid v=t ] = t\) for all \(t\), where \(y\) is the observed outcome and \(v\) is the prediction. This notion, central to algorithmic fairness, comes with a host of challenges: one of them being to assess whether a given predictor is indeed calibrated, and quantifying by how much it deviates from it. Following work by Błasiok, Gopalan, Hu, and Nakkiran which introduced a notion of distance to calibration, the paper defines the (property testing) task of calibration testing, with connections to distribution testing, and provides subquadratic-time algorithms (in the sample complexity) for the task. The authors also obtain analogous results for tolerant calibration testing, which they also introduce.

The role of shared randomness in quantum state certification with unentangled measurements, by Jayadev Acharya and Yuhan Liu (arXiv). In this paper (from January), the authors consider the following question, the quantum analogue of identity testing from the classical distribution testing world: what is the copy complexity (≈sample complexity) of certifying (≈testing) whether an unknown quantum state (≈quantum analogue of a probability distribution) is equal to a known, reference quantum state? And, crucially, what about doing this when our quantum hands are tied, i.e., without using entanglement — but possibly with adaptive measurements? This is not a new question, and we previously covered a couple papers on this in April 2020 and Feb 2021. What is new here is that the authors show it’s not about adaptivity! Mirroring what happens in the classical (distributed) case, the key here turns out to be shared randomness: that is, whether the measurements are made independently (in which case \(\Theta(d^2)\) copies are necessary and sufficient), or chosen randomly but jointly (in which case the copy complexity is \(\Theta(d^{3/2})\)).

Low Acceptance Agreement Tests via Bounded-Degree Symplectic HDXs, by Yotam Dikstein, Irit Dinur, and Alexander Lubotzky (ECCC) and Constant Degree Direct Product Testers with Small Soundness, by Mitali Bafna, Noam Lifshitz, Dor Minzer (ECCC). [Two independent works]

Let \(X\) be a (small) set of \(k\)-element subsets of \([n]\), and \(\{f_S\colon S\to \Sigma\}_{S\in X}\) a family of partial functions. Is there a way to “stitch together” all the functions \(f_S\) into a global one \(G\colon X \to \Sigma\)? A testing algorithm for this is called an agreement test, and the most natural goes as follows: pick \(S,T\in X\) at random (say, with fixed, small intersection), and accept if, and only if, \(f_{S}, f_T\) agree on \(S\cap T\). Does this work? In which parameter regime (i.e., how does the acceptance probability \(\varepsilon\) relate to the closeness-to-a-global-function-\(G\)? How large does \(X\) need to be? The two papers both show that the above agreement test works in the small soundness regime (small \(\varepsilon\)), for \(= O(n)\). Or, as the authors of the first of the two papers put it: “In words, we show that ε agreement implies global structure”

Efficient learning of quantum states prepared with few fermionic non-Gaussian gates, by Antonio Anna Mele and Yaroslav Herasymenko (arXiv). While most of the paper’s focus is on tomography (learning) of a specific class of quantum states, the authors also provide in Appendix A an algorithm for a property testing question: namely, testing the Gaussian dimension of a quantum state: specifically, tolerant testing of \(t\)-compressible \(n\)-qubit Gaussian states in trace distance (Theorem 48). I do not fully grasp what all this means, to be honest, so I’ll stop here.

Welcome to the posting for first month of 2024! We hope you had a good start to this year. Last month, we had 3 papers. So, without further delay, let’s get started.

On locally-characterized expander graphs (a survey) by Oded Goldreich (ECCC) In this paper, written in an expository style, Goldreich surveys the result of Adler, Kohler and Peng which we covered in our posts on May 2021 and August 2020. The survey begins by reminding the reader what a locally-characterizable graph property is. A family of graphs \(\mathcal{G}\) is said to be characterized by a finite class \(\mathcal{F}\) of graphs if every graph \(G \in \mathcal{G}\) is \(F\)-free for all \(F \in \mathcal{G}\). Thanks to its connections with expressibility in first order logic, one would expect the a locally-characterizable graph property to admit testers depending only on the proximity parameter (in the bounded degree graph model). So, it was quite a surprise when Adler, Kohler and Peng showed that there are locally characterizable graph properties which provably require testers whose query complexity increases with the size of the graph. The main theorem of Adler, Kohler and Peng describes the locally characterizable property that is hard to test. This characterization asserts that you can get your hands on a finite class \(\mathcal{F}\) of graphs so that \(\mathcal{F}\)-freeness of a graph \(G\) means that the graph is a (bounded-degree) expander. One of the key ingredients used in this proof is the Zig-Zag construction of Reingold, Vadhan and Wigderson.

Fast Parallel Sampling under Isoperimetry by Nima Anari, Sinho Chewi, and Thuy-Duong Vuong (arXiv) The featured paper considers the task of sampling (in parallel) from a continuous distribution \(\pi\) supported over \(\mathbb{R}^d\). The main result in the paper shows that for distributions which satisfy a log-Sobolev inequality, you can use parallelized Langevin Algorithms and obtain samples from a distribution \(\pi’\) where \(\pi’\) and \(\pi\) are close in KL-divergence. As an application of their techniques, the authors show that their results can be used to do obtain RNC samples for directed Eulerian Tours and asymmetric Determinantal Point Processes.

Holey graphs: very large Betti numbers are testable by Dániel Szabó, Simon Apers (arXiv) This paper considers the problem of testing whether an input graph \(G\) has a very large \(k\)-th Betti Number (at least \((1-\delta) d_k\) where \(d_k\) denotes the number of \(k\)-cliques in \(G\) and \(\delta > 0\) is sufficiently small) in the dense graph model. As the title indicates, the result says that this property is testable for constant \(k\). Earlier in 2010, Elek investigated this question in the bounded degree model. Elek’s main result showed that with a number of queries \(q = q(\varepsilon\), you can obtain an estimate to the \(k\)-th Betti Number which is within an additive \(\pm \varepsilon n\) of the true \(k\)-th Betti Number. Let us contrast this result with the setting of dense graphs. The authors note that in the dense graph model, getting an estimate to the \(k\)-th Betti Number which is within an additive $\pm \varepsilon d_k$ of the true value needs \(\Omega(n)\) queries. This is the reason why the authors consider the formulation above.

Happy New Year! This post covers the final property testing results for 2023. We have just two three papers, one on group testing, a new bound on sample-based testers, and a distribution testing result over higher dimensions.

On Testing Group Properties by Oded Goldreich and Laliv Tauber (ECCC). Group testing is a classic problem going back to the early days of property testing. We are given access to a “multiplication table” \(f: [n]^2 \to [n]\). Each element in \([n]\) is (supposedly) an element of a group, and \(f(i,j)\) is the product of elements \(i\) and \(j\). Our aim is to determine, in the property testing sense, whether \(f\) is the multiplication table of a group. The earliest result, from the seminal Spot-checkers paper, gave an \(\widetilde{O}(n^{3/2}/\varepsilon)\) time tester (where \(\varepsilon\) is the proximity parameter). This paper significantly improves that classic bound with a one-sided \(\widetilde{O}(n/\varepsilon)\) tester. Moreover, this tester can be adapted for testing of \(f\) is Abelian. The result is obtained by a series of testers, starting from extremely simple (yet time inefficient) to more complex versions. The best known lower bound is just \(\Omega(\log n)\), and that leaves a tantalizing (and wide open) gap to be reduced.

A Structural Theorem for Local Algorithms with Applications to Coding, Testing, and Verification by Marcel Dall’Agnol, Tom Gur, and Oded Lachish (arXiv). One of the powers of property testers is their choice of query. This is unlike the typical setting in learning, where one only gets access to random samples (or evaluations). Sample-based testers were defined to bridge this gap; so it is a property tester that only has access to (evaluations of) uniform random domain points. A sample-based tester simply checks if the sample is consistent with the property. It is natural to ask if the existence of a (vanilla) property tester implies the existence of a sample based tester. For the simplest setting, consider a \(q\)-query non-adaptive tester for some property \(\mathcal{P}\). One can visualize this as a collection of query sets of size \(q\) in a universe of size \(n\) (the domain). Naively, one might hope that with a \(q\)-way collision argument, a random sample of size \(O(n^{1-1/q})\) would contain a query set, yielding a sample based tester. Previous work showed that, for any property with a \(q\)-query non-adaptive tester, there is a sample-based tester with complexity \(O(n^{1-1/(q^2\log q)})\). Remarkably, this work gives such a bound for even adaptive testers (the best previous bound was \(O(n^{1-1/\exp(q)})\)). The result is placed in a broader framework of robust local algorithms, which subsume \(q\) query property testers, locally decodable codes (LDC), and MA proofs of proximity.

Testing Closeness of Multivariate Distributions via Ramsey Theory by Ilias Diakonikolas, Daniel M. Kane, and Sihan Liu (arXiv). (Missed from last month. -Ed) This paper considers distribution testing where the universe is \(\mathbb{R}^d\). The notion of closeness is called \(\mathcal{A}_k\) distance: we cover the universe with \(k\) axis-parallel rectangles, and “reduce” the distribution to a discrete universe of size \(k\). We then take TV-distance over these reduced distributions. When \(d=1\), the complexity of testing closeness was known to be \(\Theta(k^{4/5}/poly(\epsilon))\). For \(d > 1\), this paper gives the first non-trivial closeness testing result. The (optimal in \(k\)) bound achieved is \(\Theta(k^{6/7}/poly(\epsilon))\). Interestingly, there is a jump in the exponent on \(k\) from \(d=1\) to \(d=2\), but no jump for larger \(d\).