We saw quite an activity last month, with a total of 9 papers on property testing or related areas.

Let’s start with a paper on PCPs:
Revisiting Alphabet Reduction in Dinur’s PCP, by Venkatesan Guruswami, Jakub Opršal, and Sai Sandeep (ECCC). As mentioned in the title, this work provides an alternate proof of one of the two parts of the Dinur’s PCP theorem proof. In that proof, the argument goes by alternating two main steps: (i) amplifying the soundness gap, at the price of increasing the alphabet size (and the instance size); (ii) reducing the alphabet size, at the price of increasing the instance size. From the observation that step (i) creates some structure in the instances that can be leveraged, the authors provide an alternative construction for (ii), leading to an arguably simpler analysis of the soundness of the underlying test. A nice throwback to some of the roots of property testing.

Then, four papers on distribution testing:
Testing Mixtures of Discrete Distributions, by Maryam Aliakbarpour, Ravi Kumar, Ronitt Rubinfeld (arXiv). One of the most studied problems in distribution testing is identity testing (a.k.a., goodness-of-fit): given a reference distribution \(q\) over a domain of size \(n\) and i.i.d. samples from an unknown \(p\), decide between \(p=q\) and \(d_\textrm{TV}(p,q) > \varepsilon\). While this is now well-understood and can be done with a strongly sublinear number of samples (namely, \(\sqrt{n})\)), the case of tolerant testing is much sadder: Valiant and Valiant proved that \(\Theta(n/\log n)\) samples were required for distinguishing \(d_\textrm{TV}(p,q) < \varepsilon/2\) from \(d_\textrm{TV}(p,q) > \varepsilon\). So, noise-tolerance is almost as hard as it gets… except, as this works suggests and studies, if one has some guarantees on the noise! What if the noise in the completeness case was just that \(q\) is perturbed by some (known, or available via samples as well) noise coming from a second distribution \(\eta\)? I.e., the question is now to test whether \(p\) if of the form \(\alpha q + (1-\alpha) \eta\) for some \(\alpha\in [0,1]\), or if it is \(\varepsilon\)-far from all mixtures of \(q\) and \(\eta\). The authors have various results on this question and some extensions, but the upshot is: “with structured noise, strongly sublinear sample complexity is once again possible.”

Can Distributed Uniformity Testing Be Local?, by Uri Meir, Dor Mintzer, and Rotem Oshman (ACM PODC Proceedings). More on identity testing, specifically uniformity testing. No noise here, but another constraint: the samples are distributed. There are \(k\) players, each holding \(q\) i.i.d. samples from a distribution \(p\) over \([n]\); each can send \(\ell=1\) bit to a central referee, in a non-interactive fashion. Is \(p\) the uniform distribution, or is it \(\varepsilon\)-far from it? The authors here consider the case where \(k,n,\varepsilon\) are fixed, and prove lower bounds on the number \(q=q(k,n,\varepsilon)\) of samples each player must hold in order to perform this uniformity testing task. (Note: their lower bounds hold even in the public-randomness setting.) Motivated by the LOCAL model, they focus on 3 cases of interest, for which they obtain tight or near-tight bounds on \(q\) (in view of upper bounds from previous work of a subset of the authors): (1) when the referee’s decision has to be the \(\textsf{AND}\) of all \(n\) bits she receives; (2) when the referee can do something more involved, and use a threshold function; and (3) when the referee can use an arbitrary function of those \(n\) messages. Underlying those lower bounds is a neat application of Boolean Fourier analysis, recasting the analysis of the standard “Paninski” lower bound instance and the player’s decision functions as a question on Boolean functions.

Domain Compression and its Application to Randomness-Optimal Distributed Goodness-of-Fit, by Jayadev Acharya, Clément Canonne, Yanjun Han, Ziteng Sun, and Himanshu Tyagi (arXiv,ECCC). Remember the paper just above? Now, focus on the case (3), where the referee can use any function of the message, allow each player to send \(\ell\) bits instead of one, but give only one sample (instead of \(q\)) to each player. This becomes a setting we have talked about before on this blog (specifically, with three papers on these three months). Those three papers established the optimal number of player \(k\), in terms of the domain size \(n\), the distance parameter \(\varepsilon\), and the number of bits per player \(\ell\), in both the private-coin and public-coin settings. This new work interpolates between those two extremes, and gives the tight answer to the general question: if there are \( 0\leq s \leq \log n\) bits of public randomness available, what is the optimal number of players to perform identity testing? Behind the upper bounds is a new notion of (randomness-efficient) domain compression they introduce: how to reduce the domain size of the probability distributions while preserving the \(\ell_1\) distances between them as much as possible?

Towards Testing Monotonicity of Distributions Over General Posets, by Maryam Aliakbarpour, Themis Gouleakis, John Peebles, Ronitt Rubinfeld, and Anak Yodpinyanee (arXiv). Away from identity testing, let’s now consider another central question, testing monotonicity of distributions. Specifically, a distribution \(p\) over a given poset \((\mathcal{X}, \prec)\) is said to be monotone if \(p(x) \leq p(y)\) whenever \(x\prec y\). While the question is fully understood for the familiar case of the line \(\mathcal{X} = \{1,2,\dots,n\}\), the case of general posets is much murkier, and more or less uncharted. This paper initiates a general study of the question, improving on previously known bounds for the matching poset and Boolean hypercube, and providing several new results and techniques for the general case, to establish both upper and lower bounds on the sample complexity.

And now… graphs!
Expansion Testing using Quantum Fast-Forwarding and Seed Sets, by Simon Apers (arXiv). This paper is concerned with (bicriteria) testing of vertex expansion of a given graph \(G=(V,E)\) in the bounded-degree graph model. Loosely speaking, given a value \(\Phi\) and query access (i.e., sampling a node u.a.r., querying the degree of a node, and querying the \(i\)-th neighbor of a node) to a graph \(G\) with maximum degree \(d=\Theta(1)\), the goal is to distinguish between (i) \(G\) has vertex expansion at most \(\Phi\) and (ii) \(G\) is \(\varepsilon\)-far from any \(G’\) with vertex expansion at most \(\Phi^2\) (simplifying and dropping some technicalities). The previous state-of-the-art was a \(\tilde{O}_{\varepsilon}(n^{1/2}/\Phi^2)\) query complexity for classical algorithms, and a \(\tilde{O}_{\varepsilon}(\min(n^{1/3}/\Phi^2),n^{1/2}/\Phi^2))\) query complexity upper bound for quantum testers. The current work improves on this by providing a \(\tilde{O}_{\varepsilon}(n^{1/3}/\Phi)\) quantum tester. Graphs expand—and so does the gap between classical and quantum testing.

Walking Randomly, Massively, and Efficiently, by Jakub Łącki, Slobodan Mitrović, Krzysztof Onak, and Piotr Sankowski (arXiv). One very useful primitive, in many graph property testing results and of course well beyond property testing, is the ability to perform random walks on graphs. In the case of large, distributed (or merely impractically massive) graphs, however, it is not clear how to implement this primitive efficiently. This work addresses this question in the MPC (Massive Parallel Computation) model, and shows how to perform independently, from all of the \(n\) nodes of a graph (i.e., machine), an \(\ell\)-length random walk with only \(O(\log \ell)\) rounds of communication and \(O(n^\alpha)\) space per machine, for any constant \(\alpha \in(0,1)\). This round complexity matches a known (conditional) lower bound, and thus the result is as good as it gets — further, the authors obtain such MPC algorithms for both undirected and directed graph, somehow leveraging the latter case to handle the (significantly more complicated) directed case. As an application of this efficient random walk primitive, they provide MPC property testing algorithms for both bipartiteness and vertex expansion (same definition as in the paper above: vertex expansion \(\Phi\) vs. \(\varepsilon\)-far from any \(G’\) with vertex expansion \(\Phi^2\)) in the general graph model. (Beyond property testing, another application—the main one in the paper— is computing PageRank in both graphs and directed graphs.)

A Lower Bound on Cycle-Finding in Sparse Digraphs, by Xi Chen, Tim Randolph, Rocco Servedio, and Timothy Sun (arXiv). In this paper, the authors tackle the problem of one-sided testing of acyclicity of directed graphs, in the bounded-degree graph model (equivalently, on the task of finding a cycle in a digraph promised to be far from acyclic). Previously, an \(\Omega(\sqrt{n})\) lower bound for this task had been shown by Bender and Ron, based on a birthday-paradox-type argument; this work improves on this hardness result, showing that \(\tilde{\Omega}(n^{5/9})\) queries are necessary. Interestingly, whether achieving any \(o(n)\) query complexity is possible remains open.

Constant-Time Dynamic \((\Delta+1)\)-Coloring and Weight Approximation for Minimum Spanning Forest: Dynamic Algorithms Meet Property Testing, by Monika Henzinger and Pan Peng (arXiv). Finally, the last paper covered this month is concerned with dynamic graph algorithms: where updates to a graph (which can be both edge additions and deletions) come sequentially, and the objective is to maintain, e.g., a coloring, or an approximation of some function of the graph, at any time using as little time per update. This paper advocates the use of techniques from property testing (loosely speaking, as their locality and query-efficiency translates to fast update times) to design better dynamic graph algorithms. It illustrates this idea by obtaining several new results, in particular an (amortized) \(O(1)\)-update-time randomized algorithm to maintain a \((\Delta+1)\)-coloring in a graph with degree bound \(\Delta\). The algorithm relies on a random ranking of the nodes which, combined with a suitable randomized update rule for recoloring a vertex when needed, ensure that the recolorings will not (with high probability) “propagate” to much. This is the first time that this idea, previously used in several testing and sublinear-time papers, is used in the context of dynamic graph algorithms—adding an edge between the two areas, hopefully only the first of many.

Update: we did miss a paper! Make it ten in July…
Nearly optimal edge estimation with independent set queries, by Xi Chen, Amit Levi, Erik Waingarten (arXiv). Consider the following type of access to an unknown graph \(G=(V,E)\) on \(n=|V|\) nodes: on query \(S\subseteq V\), the oracle returns 1 if \(S\) is is an independent set, and 0 otherwise. (This sounds quite a bit like what is allowed in the group testing literature.) How many such queries are necessary to get a multiplicative estimate of the number of edges, \(m=|E|\)? In this paper, the authors improve on the previous bound of \(\min(\sqrt{m}, n^2/m)\) (ignoring logarithmic factors), obtaining an \(\min(\sqrt{m}, n/m)\) upper bound — complemented by a (nearly, up to those pesky log factors) matching lower bound.

If we missed a paper this month, or represented some result above, please mention it in the comments.

We’ve got four papers this month. A mix of distribution testing, matrix problems, and a different paper on the power of sampling.

On the Complexity of Estimating the Effective Support Size, by Oded Goldreich (ECCC). In distribution testing, a classic problem is that of approximating the support size. By a (now) classic result of Valiant-Valiant, the complexity of this problem is \(\Theta(n/\log n)\). This paper raises the question of approximating the “effective” support, and that too with more than just samples from the distribution. The \(\epsilon\)-support of discrete distribution \(\mathcal{D}\) is the smallest support among any distribution \(\mathcal{D}’\) such that \(\|\mathcal{D} – \mathcal{D}’\|_1 \leq \epsilon\) (or TV-distance). Denote this as \(supp_\epsilon(\mathcal{D})\). One can also consider a bicriteria version. Given approximation parameter \(f\) and thresholds \(\epsilon_1 < \epsilon_2\), we need to provide a number in the range \([supp_{\epsilon_2}(\mathcal{D}), f\cdot supp_{\epsilon_1}(\mathcal{D})]\). The primary model studied allows for random samples and evaluation queries (where one gets the probability of any known element of the domain). In this model, for arbitrary \(\epsilon_1, \epsilon_2\), there is a continuum of algorithms, trading off query complexity with approximation. At one end, for \(f = O(\log n)\), the query complexity is \(\widetilde{O}(1/\epsilon_1)\). At the other end, for \(f=1\), the query complexity is \(O(\log^* n)\). (Here, \(n = supp_{\epsilon_1}(\mathcal{D})\).) There are lower bounds showing the necessity of evaluation queries for subpolynomial query complexities.

Communication and Memory Efficient Testing of Discrete Distributions, by Ilias Diakonikolas, Themis Gouleakis, Daniel M. Kane, and Sankeerth Rao (arXiv). This paper adds additional computational constraints to the distribution testing problem. In the streaming setting, the algorithm is only allowed a single pass over the random samples. It has \(m\) bits of storage, much smaller than \(n\), the distribution size. The aim is to minimize the number of samples required for uniformity (and closeness) testing. For uniformity testing in the streaming setting, the paper gives an algorithm that uses \(\widetilde{O}(m + n/m)\) samples. The standard collision algorithm requires \(\Theta(\sqrt{n})\) storage (to store the samples), while this result gives a non-trivial bound for \(m \ll \sqrt{n}\). There are lower bounds showing that these bounds are basically tight. In the distributed distribution testing problem, there are a number of processors, each holding \(\ell\) samples. A referee asks a question to the processor, whose answers are broadcast to everyone. The aim is to minimize communication cost. For uniformity testing in this setting, there is a protocol using \(O(\sqrt{n\log n/\ell})\) bits of communication. As a sanity check, note that for \(\ell = \sqrt{n}\), one processor could simply run the collision algorithm locally to report the result.

Querying a Matrix through Matrix-Vector Products by Xiaoming Sun, David P. Woodruff, Guang Yang, and Jialin Zhang (arXiv). Consider \(n \times d\) matrix \(M\) over some field \(\mathbb{F}\). One gets access to this matrix through matrix-vector products, and wishes to test some matrix property. This is a natural model in many settings, and generalizes the classic setting of query access. A subtle point is that one can only right multiply with “query” vectors, and there are problems where left multiplication can change the complexity. (A nice example in the paper is testing if a square matrix is symmetric. With both left and right multiplications, this is easy, since we can directly access rows and columns. By only accessing columns, this is non-trivial.) This paper studies a number of problems, broadly classified as linear algebra problems, statistics problems, and graph problems. Some highlights are: testing symmetry can be done in \(O(1)\) queries, and the maximum eigenvalue can be approximated in \(O(\log n)\) queries adaptively (but there is an \(\Omega(n)\) non-adaptive lower bound). For graph problems, here’s an interesting discovery. If \(M\) is the adjacency matrix, connectivity requires \(\Omega(n/\log n)\) queries. But if \(M\) is the signed edge-vertex matrix, then this can be done in \(poly(\log n)\) queries. This paper provides a number of interesting directions and problems to study.

The Adversarial Robustness of Sampling by Omri Ben-Eliezer and Eylon Yogev (arXiv). This isn’t a property testing paper, but how can one ignore a paper on understanding the power of sampling? It’s convenient to think of the following setting. An algorithm gets a stream of \(n\) numbers, and has to answer some questions about the stream (say, the median or other quantiles). The simplest strategy is to take a small random sample of the stream. But what if an adversary was generating the stream, depending on the current sample? Under what circumstances is the sample still “representative” of the stream? The specific results of this paper require getting into set systems and VC dimension, which I’ll leave for the sake of simplicity. Let’s go back to the median question. To get an approximate median, a constant number of samples suffice. A consequence of the main result is that if one takes \(O(\log n)\) samples, then this is robust to adversarial streams. On the other hand, lower bounds show that constant sized samples can be fooled by an adversary. The paper is a really interesting read, and is a nice take on “standard facts” that we take for granted.

We were able to find four new papers for May 2019 — as usual, please let us know if we missed any!
EDIT: We did, in fact, miss one paper, which is the bottom one listed below.

On Local Testability in the Non-Signaling Setting, by Alessandro Chiesa, Peter Manohar, and Igor Shinkar (ECCC). This paper studies testability of a certain generalization of (distributions over) functions, known as \(k\)-non-signalling functions, objects which see use in hardness of approximation and delegation of computation. Prior work by the authors show the effectiveness of the linearity test in this setting, leading to the design of PCPs. On the other hand, in this work, the authors show that two types of bivariate tests are ineffective in revealing low-degree structure of these objects.

Computing and Testing Small Vertex Connectivity in Near-Linear Time and Queries, by Danupon Nanongkai, Thatchaphol Saranurak, and Sorrachai Yingchareonthawornchai (arXiv). This work, apparently simultaneous with the one by Forster and Yang that we covered last month, also studies the problem of locally computing cuts in a graph. The authors also go further, and study approximation algorithms for the same problems. Inspired by the connections to property testing in the work of Forster and Yang, they apply these approximation algorithms to get even more query-efficient algorithms for the problems of testing \(k\)-edge- and \(k\)-vertex-connectivity.

Testing Graphs against an Unknown Distribution, by Lior Gishboliner and Asaf Shapira (arXiv). This paper studies graph property testing, under the vertex-distribution-free (VDF) model, as recently introduced by Goldreich. In the VDF model, rather than the ability to sample a random node, the algorithm has the ability to sample a node from some unknown distribution, and must be accurate with respect to the same distribution (reminiscent of the PAC learning model). In Goldreich’s work, it was shown that every property which is testable in the VDF model is semi-hereditary. This work strengthens this statement and proves a converse, thus providing a characterization: a property is testable in the VDF model if and only if it is both hereditary and extendable. These descriptors roughly mean that the property is closed under both removal and addition of nodes (with the choice of addition of edges in the latter case). This is a far simpler characterization than that of properties which are testable in the standard model, which is a special case of the VDF model.

Private Identity Testing for High-Dimensional Distributions, by Clément L. Canonne, Gautam Kamath, Audra McMillan, Jonathan Ullman, and Lydia Zakynthinou (arXiv). This work continues a recent line on distribution testing under the constraint of differential privacy. The settings of interest are multivariate distributions: namely, product distributions over the hypercube and Gaussians with identity covariance. An application of a statistic of CDKS, combined with a Lipschitz extension from the set of datasets likely to be generated by such structured distributions, gives a sample-efficient algorithm. A time-efficient version of this extension is also provided, at the cost of some loss in the sample complexity. Some tools of independent interest include reductions between Gaussian mean and product uniformity testing, balanced product identity to product uniformity testing, and an equivalence between univariate and “extreme” product identity testing.

Testing Bipartitness in an Augmented VDF Bounded-Degree Graph Model, by Oded Goldreich (arXiv). Another work on the vertex-distribution-free (VDF) model, as described above. In this one, Goldreich considers an augmentation of the model, where the algorithm is further allowed to query the probability of each node. With this augmentation, he gives \(\tilde O(\sqrt{n})\)-time algorithms for testing bipartiteness and cycle-free-ness, where \(n\) is the “effective support” of the distribution. That is, \(n\) is the number of nodes in the graph after discarding the nodes with minimal probability until \(\varepsilon/5\) mass is removed.

After a quite slow month of March, things sped up quite significantly in April: six different papers, ranging from graph testing to function testing to quantum distribution testing!

Update (05/04): We missed one. Seven!

Junta correlation is testable, by Anindya De, Elchanan Mossel, and Joe Neeman (arXiv). Junta testing really has seen a lot of attention and progress over the pass couple years! In this new work, the focus is on the tolerant version of (Boolean) junta testing, where the goal is to decide whether a given function \(f\colon \{-1,1\}^n\to \{-1,1\}\) is close to some \(k\)-junta, or far from any such simple function. The current paper improves on previous work of Blais, Canonne, Eden, Levi, and Ron, which showed how to distinguish between \(\varepsilon\)-close to \(k\)-junta and \(16\varepsilon\)-far from \(k\)-junta in \(O_{k,\varepsilon}(1)\) queries, in two ways: (i) the gap-factor of 16 can now be made arbitrarily small, yielding the first fully tolerant non-trivial junta tester; and (ii) the new algorithm also identifies the underlying “core junta” (up to permutation of the coordinates). Besides the other results contained in the paper (such as results for a relaxation of the tolerant question akin to that of Blais et al.), a key aspect of this paper is to go beyond the use of (set) influence as a proxy for distance to junta-ness which underlied all previous work, thus potentially opening a new avenue towards get fully tolerant, \(\mathrm{poly}(k,1/\varepsilon)\)-query junta testing algorithms.

Testing Tensor Products, by Irit Dinur and Konstantin Golubev (arXiv). In this paper, the authors consider the following testing question: given query access to a Boolean function \(f\colon [n]^d \to \mathbb{F}_2\), test whether \(f\) is of the form \(f(x) = f_1(x_1)+\dots f_d(x_d)\) (equivalently, this is the task of testing whether a \(d\)-dimensional tensor with \(\pm 1\) is of rank \(1\)). They provide two different proximity-oblivious testers (POT) for this task: a \(4\)-query one, reminiscent and building upon the BLR linearity test; as well as a \((d+1)\)-query POT (which they dub “Shapka Test,” one can assume for their love of warm and comfortable hats), whose analysis is significantly simpler.

Changing direction, the next paper we saw is concerned with quantum testing of (regular, good ol’ classical) probability distributions:

Quantum Algorithms for Classical Probability Distributions, by Aleksandrs Belovs (arXiv). This work on quantum distribution testing considers 4 of the existing models of access to samples from an unknown probability distribution, analyzes their relationship, and argues their merits. Further, the autho establishes that for the simple hypothesis testing question of distinguishing between two (known, fixed) probability distributions \(p,q\), in all four models the optimal sample complexity is proportional to the inverse of the Hellinger distance between \(p\) and \(q\)—in contrast to the classical setting, where it is known to be proportional to the squared inverse of this distance.

Turning to graphs, the next work considers clustering of bounded-degree graphs, a topic we recently discussed as well:

Robust Clustering Oracle and Local Reconstructor of Cluster Structure of Graphs, by Pan Peng (arXiv). Consider the following noisy model: an unknown bounded-degree graph (of maximum degree \(d\)) \(G\) which is \((k, \phi_{\rm in},\phi_{\rm out})\)-clusterable (i.e., clusterable in at most \(k\) clusters of inner and outer conductances bounded by \(\phi_{\rm in},\phi_{\rm out}\)) is adversarially modified in at most \($\varepsilon d n\) edges between clusters. This noise model, introduced in this work, leads to the natural questions of “recovering the underlying graph \(G\)”: this is what the author tackles, by designing sublinear-time local clustering oracles and local reconstruction algorithms (local filters) in this setting. Further, in view of the noise model reminiscent of property testing in the bounded-degree graph setting, connections to testing clusterability are discussed; the implications of the results for testing clusterability are discussed in Section 1.4.

A Faster Local Algorithm for Detecting Bounded-Size Cuts with Applications to Higher-Connectivity Problems, by Sebastian Forster and Liu Yang (arXiv). The authors focus on the problem of finding an edge (or vertex) cut in a given graph from a local viewpoint, i.e., in the setting of local computation algorithms, and provide a slew of results in detecting bounded-size cuts. This in turn has direct implications for testing \(k\)-edge and \(k\)-vertex connectivity, directed and undirected, in both the bounded-degree and general (unbounded degree) models, improving on or subsuming the current state-of-the-art across the board (see Section 1.2.4 of the paper, and the very helpful Tables 3 and 4, for a summary of the improvements).

Random walks and forbidden minors II: A \(\mathrm{poly}(d\varepsilon^−1)\)-query tester for minor-closed properties of bounded degree graphs, by Akash Kumar, C. Seshadhri, and Andrew Stolman (ECCC). Following their breakthrough from last May, the authors are at it again! Leveraging a random-walk approach, they resolve the following open question in testing bounded-degree graph: is there a \(\mathrm{poly}(1/\varepsilon)\)-query tester for planarity (and, more generality, for minor-closed graph properties)? The previous state-of-the-art, due to Levi and Ron, had a quasipolynomial dependence on \(1/\varepsilon\).
Without spoiling too much: the answer to the open question is yes— and further the authors settle it by showing an even more general result on testing \(H\)-freeness (for any constant-size graph \(H\)).

Update (05/04):

Testing Unateness Nearly Optimally, by Xi Chen and Erik Waingarten (arXiv). Recall that a function \(f\colon \{0,1\}^n\to \{0,1\}\) is said to be unate if there exists some \(s\in\{0,1\}^n\) such that \(f(\cdot \oplus s)\) is monotone; i.e., if \(f\) is either non-increasing or non-decreasing in each coordinate. Testing unateness has seen a surge of interest over the past year or so; this work essentially settles the question, up to polylogarithmic factors in \(n\) (and the exact dependence on \(\varepsilon\)). Namely, the authors present and analyze an \(\tilde{O}(n^{2/3}/\varepsilon^2)\)-query adaptive tester for unateness, which nearly matches the \(\tilde{\Omega}(n^{2/3})\)-query lower bound for adaptive testers previously established by Chen, Waingarten, and Xie.

Last month was very calm. Eerily calm in fact: no property testing or related sublinear-time algorithm in view!* Gird your loins for the April batch, I reckon?

\({}^\ast\) That we saw, at least. if we missed some, please mention it in the comments below!)

Update: As mentioned in the comments, we did indeed missed two (related) works on distribution estimation from a competitive viewpoint. Namely, for a large class of properties (entropy, distance to uniformity, support size…), Hao, Orlitsky, Suresh, and Wu provide in the first paper (arXiv 1904.00070) an “instance-optimal(ish)” estimator which does “as well” with \(m/\sqrt{\log m}\) samples than the natural and naive empirical estimator would do with \(m\) (thus, in some sense, amplifying the data size). In the following paper (arXiv 1903.01432), Hao and Orlitsky improve this and remove the “ish” to get the optimal amplification \(m/\log m\).

A lean month by our (now high) standards. Three papers, on local computational algorithms for spanners, sublinear set cover, and quantum distribution testing.

Local Computation Algorithms for Spanners, by Merav Parter, Ronitt Rubinfeld, Ali Vakilian, Anak Yodpinyanee (arXiv). Consider the problem of computing a \(k\)-spanner for a graph \(G\). This is a subgraph \(H\), where (for any pair of vertices) the shortest path distance in \(H\) is at most \(k\) times the corresponding distance in \(G\). A Local Computation Algorithm (LCA) essentially gives sublinear query access to (some spanner) \(H\), without any preprocessing on \(G\). In other words, an LCA takes as input an edge \((i,j)\) in \(G\), and in sublinear time says “yes” or “no”. Despite no preprocessing, it is guaranteed that all the “yes” answers form a spanner, and the number of “yes” answers is small. One of the main challenges in LCAs for graph problems is getting a polynomial dependence on \(\Delta\), the maximum degree. (Many results end up with an exponential dependence on \(\Delta\), essentially assuming bounded-degree.) This paper give an LCA outputting spanners of optimal size for \(k=3,5\), with query complexity \(n^{1-\alpha}\) (for constant \(\alpha\) depending on \(k\)). The second result works for general \(k\), but has a query complexity that is \(O(\Delta^4n^{2/3})\).

Set Cover in Sub-linear Time, by Piotr Indyk, Sepideh Mahabadi, Ronitt Rubinfeld, Ali Vakilian, Anak Yodpinyanee (arXiv). Set cover needs no introduction. This paper considers sublinear algorithms for set cover. It is convenient to think of the instance as a binary \(m \times n\) matrix, for \(m\) sets over a universe of size \(n\). The query model gives access to the \(j\)th non-zero entry in a desired column/row. The aim is this paper is to actually construct a set cover, and not just approximate its size (the latter problem with initiated in Nguyen-Onak). The first result shows a black-box conversion of any set cover algorithm into a sublinear query complexity algorithm. Any \(\alpha\)-approximation for set cover can be converted into an \(O(\alpha)\)-approximation, that makes \(O(m(n/k)^{\beta} + nk)\) (where \(k\) is the optimal set cover size, and \(\beta < 1\) depends on \(\alpha\)). For large \(k\), the paper gives an \(O(mn/k)\) query, \(O(\log n)\)-approximation algorithm. There are various lower bounds given, that show these dependencies are necessary.

Distributional Property Testing in a Quantum World by András Gilyén and Tongyang Li (arXiv). There are two aspects to quantum distribution testing. First, we can think of faster distribution testing (for classical problems) using quantum computation. Second, we can generalize the problem of distribution testing to its quantum equivalent, called density operator testing (this papers gives the first result for this quantum generalization). A distribution is a non-negative vector in \(\mathbb{R}^n\) with unit \(l_1\)-norm. The quantum generalization, a density operator, is a PSD matrix in \(\mathbb{C}^n\) with unit trace. This paper has a number of results, along both aspects. For example, for the well-known problem of testing equality of distributions under total variation distance, the papers gives an \(\widetilde{O}(\sqrt{n}/\epsilon)\) tester (an improvement over classical lower bounds). For testing equality of density operators under the same distance, the paper gives an \(O(n/\epsilon)\) tester (note that the trivial bound is \(O(n^2)\), since the operator is a matrix).

Minimax Testing of Identity to a Reference Ergodic Markov Chain, by Geoffrey Wolfer and Aryeh Kontorovich (arXiv). This work studies distributional identity testing on Markov chains from a single trajectory, as recently introduced by Daskalakis, Dikkala, and Gravin: we wish to test whether a Markov chain is equal to some reference chain, or far from it. This improves on previous work by considering a stronger distance measure than before, and showing that the sample complexity only depends on properties of the reference chain (which we are trying to test identity to). It additionally proves instance-by-instance bounds (where the sample complexity depends on properties of the specific chain we wish to test identity to).

Almost Optimal Distribution-free Junta Testing, by Nader H. Bshouty (arXiv). This paper provides a \(\tilde O(k/\varepsilon)\)-query algorithm with two-sided error for testing if a Boolean function is a \(k\)-junta (that is, its value depends only on \(k\) of its variables) in the distribution-free model (where distance is measured with respect to an unknown distribution from which we can sample). This complexity is a quadratic improvement over the \(\tilde O(k^2)/\varepsilon\)-query algorithm of Chen, Liu, Servedio, Sheng, and Xie. This complexity is also near-optimal, as shown in a lower bound by Saglam (which we covered back in August).

Exponentially Faster Massively Parallel Maximal Matching, by Soheil Behnezhad, MohammadTaghi Hajiaghayi, and David G. Harris (arXiv). The authors consider maximal matching in the Massively Parallel Computation (MPC) model. They show that one can compute a maximal matching in \(O(\log \log \Delta)\)-rounds, with \(O(n)\) space per machine. This is an exponential improvement over the previous works, which required either \(\Omega(\log n)\) rounds or \(n^{1 + \Omega(1)}\) space per machine. Corollaries of their result include approximation algorithms for vertex cover, maximum matching, and weighted maximum matching.

Happy near year, and best wishes to those close and \(\varepsilon\)-far! December concluded the year with 4 new preprints, spanning quite a lot of the property testing landscape:

Testing Stability Properties in Graphical Hedonic Games, by Hendrik Fichtenberger and Anja Rey (arXiv). The authors of this paper consider the problem of deciding whether a given hedonic game possesses some “coalition stability” in a property testing framework. Namely, recall that a hedonic game is a game where players (nodes) form coalitions (subsets of nodes) based on their individual preferences and local information about the considered coalition, thus resulting in a partition of the original graph.
Several notions exist to evaluate how good such a partition is, based on how “stable” the given coalitions are. This work focuses on hedonic games corresponding to bounded-degree graphs, introducing and studying the property testing question of deciding (for several such notions of stability) whether a given game admits a stable coalition structure, or is far from admitting such a partition.

Spectral methods for testing cluster structure of graphs, by Sandeep Silwal and Jonathan Tidor (arXiv). Staying among bounded-degree graphs, we turn to testing clusterability of graphs, the focus of this paper. Given an \(n\)-node graph \(G\) of degree at most \(d\) and parameters \(k, \phi\), say that \(G\) is \((k, \phi)\)-clusterable if it can be partitioned in \(k\) parts of inner conductance at least \(\phi\).
Analyzing properties of a random walk on \(G\), this work gives a bicriterion guarantee (\((k, \phi)\)-clusterable vs. \(\varepsilon\)-far from \((k, \phi^\ast)\)-clusterable, where \(\phi^\ast \approx \varepsilon^2\phi^2\)) for the case \(k=2\), improving on previous work by Czumaj, Peng, and Sohler’15.

We then switch from graphs to probability distributions with our third paper:

Inference under Information Constraints I: Lower Bounds from Chi-Square Contraction, by Jayadev Acharya, Clément Canonne, and Himanshu Tyagi (arXiv). (Disclaimer: I’m one of the authors.) In this paper, the first of an announced series of three, the authors generalize the settings of two previous works we covered here and there to consider the general question of distribution testing and learning when the \(n\) i.i.d. samples are distributed among \(n\) players, which each can only communicate their sample to the central algorithm by respecting some pre-specified local information constraint (e.g., privacy, or noise, or communication budget). This paper develops a general lower bound framework to study such questions, with a systematic focus on the power of public vs. private randomness between the \(n\) parties, and instantiate it to obtain tight bounds in the aforementioned locally private and communication-limited settings. (Spoiler: public randomness strictly helps, but not always.)

Finally, after games, graphs, and distributions, our fourth paper of the month concerns testing of functions:

Partial Function Extension with Applications to Learning and Property Testing, by Umang Bhaskar and Gunjan Kumar (arXiv). This work focuses on a problem quite related to property testing, that of partial function extension: given as input \(n\) pairs point/value from a purported function on a domain \(X\) of size \(|X| > n\), one is tasked with deciding whether there does exist (resp., with finding) a function \(f\) on \(X\) consistent with these \(n\) values which further satisfies a specific property, such as linearity or convexity. This is indeed very reminiscent of property testing, where one gets to query these \(n\) points and must decide (approximate) consistency with such a well-behaved function. Here, the authors study the computational hardness of this partial function extension problem, specifically for properties such as subadditivity and XOS (a sub-property of subadditivity); and as corollaries obtain new property testers for the classes of subadditive and XOS functions.

As usual, if you know of some work we missed from last December, let us know in the comments!

Alas! Sorry for the delay, but November was a good month with five seven papers. We missed a few papers from September, so please look at the updated post for September as well. (Update: Thanks to Omri Ben-Eliezer and an anonymous poster for pointing out two missed papers.)

From Local to Robust Testing via Agreement Testing, by Irit Dinur, Tali Kaufman, and Noga Ron-Zewi (ECCC). Consider a code, which is simply a subset of \(C \subseteq \mathbb{F}^n_q\). (Typically, we think of linear codes, which form a linear subspace.) The study of locally testable codes is a rich area. Consider an input \(x \in \mathbb{F}^n_q\). A local tester for code/property \(C\) queries, according to some distribution, a subset \(K\) of \(Q\) coordinates in \(x\), and rejects if the “view” \(x|_K\) is inconsistent with any codeword. The rejection probability should be proportional to the distance of \(x\) to \(C\), denoted \(dist(x,C)\). A robust tester demands the average distance of \(x|_K\) to the corresponding property \(C|_K\) must be proportional to \(dist(x,C)\). This is a much stronger demand than that of just local testing. The main result is local testability implies robust testability for the special class of lifted codes. The proof goes via agreement testing, where an algorithm gets access to some fragments of \(x\), and must decide if these fragments are consistent with a codeword.

Testing local properties of arrays, by Omri Ben-Eliezer (ECCC). Consider a function \(f:[n]^d \to \Sigma\), where \(\Sigma\) is finite. Now, suppose we define a property \(\mathcal{P}\) with respect to a set of forbidden \([k]^d\) (consecutive) subarrays. Such a property is called \(k\)-local. The inspiration for this work is a line of research on testing of image properties. Note that for \(k=2,3\), this notion subsumes a number of properties, such as monotonicity, \(c\)-Lipschitz, submodularity, convexity, etc. The main result is a one-sided, non-adaptive tester for all such properties. Ignoring \(\epsilon, k\) dependencies, the query complexity is \(O(\log n)\) for \(d=1\) and \(O(n^{d-1})\) for \(d > 1\). Remarkably, there is no dependence on the alphabet size \(|\Sigma|\). Moreover, the bound above is optimal for one-sided, non-adaptive testers. The basic idea to pick blocks of various sizes, and query all points on the boundary. Then, the tester can determine (by sort of brute force search) if this is consistent with some function in \(\mathcal{P}\). The key is to prove that for a function that is far from \(\mathcal{P}\), there are many blocks that cannot be consistent with \(\mathcal{P}\).

Erasures vs. Errors in Local Decoding and Property Testing, by Sofya Raskhodnikova, Noga Ron-Zewi and Nithin Varma (ECCC). The main theme of this paper is the notion of erasure-resilient testing. Consider property \(\mathcal{P}\). Suppose \(\alpha\)-fraction of the coordinates of input \(x\) is hidden. Thus, queries to these coordinates of \(x\) return nothing. Our aim is to accept if there is some “filling in” of the hidden coordinates that make \(x\) satisfy \(\mathcal{P}\), and reject if for all fillings, \(x\) remains from from \(\mathcal{P}\). If \(\alpha = 0\), this is standard property testing. Suppose there exists an \((\alpha, \alpha+\epsilon)\)-tolerant tester for \(\mathcal{P}\). (I’m fudging factors a bit here for readability.) We get an \(\epsilon\)-tester with \(\alpha\)-fraction of erasures, by simply filling \(x\) in arbitrarily. So this model is in between vanilla and tolerant testing. The main result is a separation. There is a property that is constant-query testable under erasures, but requires \(n^{\Omega(1)}\) queries to tolerant test (with the above parameters). One of the main technical results is a list decoder for the Hadamard code that can tolerate erasures of any \(\alpha \lt 1\).

Domain Reduction for Monotonicity Testing: A o(d) Tester for Boolean Functions on Hypergrids, by Hadley Black, Deeparnab Chakrabarty, and C. Seshadhri (ECCC). Consider monotonicity testing of Boolean functions of the hypergrid, \(f:[n]^d \to \{0,1\}\). It is known that there are \(O(d)\) testers with complexity independent of \(n\), while for \(n=2\), there are \(o(d)\) testers. The main result is a tester with both these properties. The paper gives an \(O(d^{5/6})\) tester for such functions. The main technical ingredient is a domain reduction theorem. Consider restrictions of \(f\) to a uniform random \([poly(d\epsilon^{-1})]^d\) hypergrid. The paper proves that if \(f\) is \(\epsilon\)-far from monotone, then (with high probability) so is the restriction. Thus, for the purposes of monotonicity testing, one can simply assume that \(n = poly(d)\). This gives a black-box method to get testers independent of \(n\).

On the Testability of Graph Partition Properties, by Yonatan Nakar and Dana Ron (ECCC). Consider property testing on dense graphs. The seminar result of Goldreich-Goldwasser-Ron studies graph partitioning properties. Such a property is defined as follows. Fix constant \(k\). We wish to partition the vertices of graph \(G\) into \(V_1, V_2, \ldots, V_k\). For each \(i\), \(|V_i|/n \in [\rho^L_i, \rho^U_i]\). Furthermore, for each \(i, j\), the number of edges between \(V_i, V_j\) must be in \([\rho^L_{i,j}n^2, \rho^U_{i,j}n^2]\). (All the \(\rho\)s are parameters of the property.) This subsumes properties like \(k\)-colorability, and the classic result is a \(poly(1/\epsilon)\)-tester for these properties. Note that the edge conditions are absolute, with respect to \(n^2\), and not with respect to \(|V_i|\cdot |V_j|\). This paper allows for relative bounds of the form \([\rho^L_{i,j}|V_i|\cdot |V_j|, \rho^U_{i,j}|V_i|\cdot |V_j|]\). This is far more expressive, subsuming properties like having large cliques, or large complete bipartite graphs. One of the main results is a (two-sided) \(poly(1/\epsilon)\) tester for all such properties. Furthermore, there is a characterization of such properties that are one-sided testable in \(poly(1/\epsilon)\) queries. For such properties, it is proven that the density conditions (between \(V_i, V_j\)) must either be unconstrained or have \(\rho^L_{i,j} = \rho^U_{i,j}\) be either 1 or 0.

Limits of Ordered Graphs and Images, by Omri Ben-Eliezer, Eldar Fischer, Amit Levi, and Yuichi Yoshida (arXiv). The theory of graph limits is a deep topic, with connections to property testing, regularity lemmas, and analysis of real-world networks. Roughly speaking, consider a sequence \(\{G_n\}\) of graphs where the (relative) frequency of any fixed subgraph converges. Then, we can define a limit of the graphs, called a graphon, which is a measurable function \(W:[0,1]^2 \to [0,1]\). This paper discovers appropriate limits of ordered graphs. Note that images can be represented as ordered graphs. An ordered graph is a symmetric function \(G:[n]^2 \to \{0,1\}\) (with the trivial automorphism group). The graphon definitions and limits do not generalize here. There is a sequence of ordered graphs where the ordered subgraph frequencies converge, but one cannot associate any graphon as the limit. This paper discovered a new object called an orderon, which is a function \(W:([0,1]^2)^2 \to [0,1]\). Orderons can be shown to be the limits of sequences of ordered graphs.

Two Party Distribution Testing: Communication and Security, by Alexandr Andoni, Tal Malkin, and Negev Shekel Nosatzki (arXiv). This paper gives a new take on distribution testing. Suppose Alice and Bob have each have access to \(t\) samples from (unknown) distributions \(p\) and \(q\) respectively. Their aim is to distinguish \(p = q\) from \(\|p-q\|_{TV} > \epsilon\). How much communication is required? In the standard setting, it is known that \(n^{2/3}\) (ignoring \(\epsilon\)) is necessary and sufficient. In the communication setting, the bound turns out to basically be \(\Theta((n/t)^2)\). Observe how for \(t=n^{2/3}\), the communication is \(n^{2/3}\) (saying that Alice is simply going to send all her samples to Bob). But for larger \(t\), the communication decreases. Since Alice can learn more about her distribution, she can use sketching techniques to reduce her communication. There is an analogous result for testing independence of two distributions. Furthermore, one can even require these protocols to be secure, so Alice and Bob learn nothing about each others samples (other than the final answer).

As October draws to a close, we are left with four new papers this month.

Testing Matrix Rank, Optimally, by Maria-Florina Balcan, Yi Li, David P. Woodruff, Hongyang Zhang (arXiv). This work investigates the problem of non-adaptively testing matrix properties, in both the standard query model and the more general sensing model, in which the algorithm may query the component-wise inner product of the matrix with “sensing” matrices. It proves tight upper and lower bounds of \(\tilde \Theta(d^2/\varepsilon)\) for the query model, and eliminating the dependence on \(\varepsilon\) in the sensing model. Furthermore, they introduce a bounded entry model for testing of matrices, in which the entries have absolute value bounded by 1, in which they prove various bounds for testing stable rank, Schatten-\(p\) norms, and SVD entropy.

Testing Halfspaces over Rotation-Invariant Distributions, by Nathaniel Harms (arXiv). This paper studies the problem of testing from samples whether an unknown boolean function over the hypercube is a halfspace. The algorithm requires \(\tilde O(\sqrt{n}/\varepsilon^{7})\) random samples (which has a dependence on \(n\) which is tight up to logarithmic factors) and works for any rotation-invariant distribution, generalizing previous works that require the distribution be Gaussian or uniform.

Testing Graphs in Vertex-Distribution-Free Models, by Oded Goldreich (ECCC). While distribution-free testing has been well-studied in the context of Boolean functions, it has not been significantly studied in the context of testing graphs. In this context, distribution-free roughly means that the algorithm can sample nodes of the graph according to some unknown distribution \(D\), and must be accurate with respect to the measure assigned to nodes by the same distribution. The paper investigates various properties which may be tested with a size-independent number of queries, including relationships with the complexity of testing in the standard model.

A Theory-Based Evaluation of Nearest Neighbor Models Put Into Practice, by Hendrik Fichtenberger and Dennis Rohde (arXiv). In the \(k\)-nearest neighbor problem, we are given a set of points \(P\), and the answer to a query \(q\) is the set of the \(k\) points in \(P\) which are closest to \(q\). This paper considers the following property testing formulation of the problem: given a set of points \(P\) and a graph \(G = (P,E)\), is each point \(p \in P\) connected to its \(k\)-nearest neighbors, or is it far from being a \(k\)NN graph? The authors prove upper and lower bounds on the complexity of this problem, which are both sublinear in the number of points \(n\).