
Agreement tests on graphs and hypergraphs
Agreement tests are a generalization of low degree tests that capture a ...
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PCA of high dimensional random walks with comparison to neural network training
One technique to visualize the training of neural networks is to perform...
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Pairwise Independent Random Walks can be Slightly Unbounded
A family of problems that have been studied in the context of various st...
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SynsetRank: Degreeadjusted Random Walk for Relation Identification
In relation extraction, a key process is to obtain good detectors that f...
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KMT coupling for random walk bridges
In this paper we prove an analogue of the KomlósMajorTusnády (KMT) emb...
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Random walks and forbidden minors II: A poly(dε^1)query tester for minorclosed properties of boundeddegree graphs
Let G be a graph with n vertices and maximum degree d. Fix some minorcl...
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A Divergence Proof for Latuszynski's CounterExample Approaching Infinity with Probability "Near" One
This note is a technical supplement to the following paper: latuszynski2...
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Agreement testing theorems on layered set systems
We introduce a framework of layered subsets, and give a sufficient condition for when a set system supports an agreement test. Agreement testing is a certain type of property testing that generalizes PCP tests such as the plane vs. plane test. Previous work has shown that high dimensional expansion is useful for agreement tests. We extend these results to more general families of subsets, beyond simplicial complexes. These include  Agreement tests for set systems whose sets are faces of high dimensional expanders. Our new tests apply to all dimensions of complexes both in case of twosided expansion and in the case of onesided partite expansion. This improves and extends an earlier work of Dinur and Kaufman (FOCS 2017) and applies to matroids, and potentially many additional complexes.  Agreement tests for set systems whose sets are neighborhoods of vertices in a high dimensional expander. This family resembles the expander neighborhood family used in the gapamplification proof of the PCP theorem. This set system is quite natural yet does not sit in a simplicial complex, and demonstrates some versatility in our proof technique.  Agreement tests on families of subspaces (also known as the Grassmann poset). This extends the classical low degree agreement tests beyond the setting of low degree polynomials. Our analysis relies on a new random walk on simplicial complexes which we call the “complement random walk” and which may be of independent interest. This random walk generalizes the nonlazy random walk on a graph to higher dimensions, and has significantly better expansion than previouslystudied random walks on simplicial complexes.
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