65 research outputs found
Explicit Abelian Lifts and Quantum LDPC Codes
For an abelian group H acting on the set [?], an (H,?)-lift of a graph G? is a graph obtained by replacing each vertex by ? copies, and each edge by a matching corresponding to the action of an element of H.
Expanding graphs obtained via abelian lifts, form a key ingredient in the recent breakthrough constructions of quantum LDPC codes, (implicitly) in the fiber bundle codes by Hastings, Haah and O\u27Donnell [STOC 2021] achieving distance ??(N^{3/5}), and in those by Panteleev and Kalachev [IEEE Trans. Inf. Theory 2021] of distance ?(N/log(N)). However, both these constructions are non-explicit. In particular, the latter relies on a randomized construction of expander graphs via abelian lifts by Agarwal et al. [SIAM J. Discrete Math 2019].
In this work, we show the following explicit constructions of expanders obtained via abelian lifts. For every (transitive) abelian group H ? Sym(?), constant degree d ? 3 and ? > 0, we construct explicit d-regular expander graphs G obtained from an (H,?)-lift of a (suitable) base n-vertex expander G? with the following parameters:
ii) ?(G) ? 2?{d-1} + ?, for any lift size ? ? 2^{n^{?}} where ? = ?(d,?),
iii) ?(G) ? ? ? d, for any lift size ? ? 2^{n^{??}} for a fixed ?? > 0, when d ? d?(?), or
iv) ?(G) ? O?(?d), for lift size "exactly" ? = 2^{?(n)}. As corollaries, we obtain explicit quantum lifted product codes of Panteleev and Kalachev of almost linear distance (and also in a wide range of parameters) and explicit classical quasi-cyclic LDPC codes with wide range of circulant sizes.
Items (i) and (ii) above are obtained by extending the techniques of Mohanty, O\u27Donnell and Paredes [STOC 2020] for 2-lifts to much larger abelian lift sizes (as a byproduct simplifying their construction). This is done by providing a new encoding of special walks arising in the trace power method, carefully "compressing" depth-first search traversals. Result (iii) is via a simpler proof of Agarwal et al. [SIAM J. Discrete Math 2019] at the expense of polylog factors in the expansion
Spectra of lifted Ramanujan graphs
A random -lift of a base graph is its cover graph on the vertices
, where for each edge in there is an independent
uniform bijection , and has all edges of the form .
A main motivation for studying lifts is understanding Ramanujan graphs, and
namely whether typical covers of such a graph are also Ramanujan.
Let be a graph with largest eigenvalue and let be the
spectral radius of its universal cover. Friedman (2003) proved that every "new"
eigenvalue of a random lift of is with high
probability, and conjectured a bound of , which would be tight by
results of Lubotzky and Greenberg (1995). Linial and Puder (2008) improved
Friedman's bound to . For -regular graphs,
where and , this translates to a bound of
, compared to the conjectured .
Here we analyze the spectrum of a random -lift of a -regular graph
whose nontrivial eigenvalues are all at most in absolute value. We
show that with high probability the absolute value of every nontrivial
eigenvalue of the lift is . This result is
tight up to a logarithmic factor, and for it
substantially improves the above upper bounds of Friedman and of Linial and
Puder. In particular, it implies that a typical -lift of a Ramanujan graph
is nearly Ramanujan.Comment: 34 pages, 4 figure
Gradient Coding from Cyclic MDS Codes and Expander Graphs
Gradient coding is a technique for straggler mitigation in distributed
learning. In this paper we design novel gradient codes using tools from
classical coding theory, namely, cyclic MDS codes, which compare favorably with
existing solutions, both in the applicable range of parameters and in the
complexity of the involved algorithms. Second, we introduce an approximate
variant of the gradient coding problem, in which we settle for approximate
gradient computation instead of the exact one. This approach enables graceful
degradation, i.e., the error of the approximate gradient is a
decreasing function of the number of stragglers. Our main result is that
normalized adjacency matrices of expander graphs yield excellent approximate
gradient codes, which enable significantly less computation compared to exact
gradient coding, and guarantee faster convergence than trivial solutions under
standard assumptions. We experimentally test our approach on Amazon EC2, and
show that the generalization error of approximate gradient coding is very close
to the full gradient while requiring significantly less computation from the
workers
Explicit near-Ramanujan graphs of every degree
For every constant and , we give a deterministic
-time algorithm that outputs a -regular graph on
vertices that is -near-Ramanujan; i.e., its eigenvalues
are bounded in magnitude by (excluding the single
trivial eigenvalue of~).Comment: 26 page
Using discrepancy to control singular values for nonnegative matrices
AbstractWe will consider two parameters which can be associated with a nonnegative matrix: the second largest singular value of the “normalized” matrix, and the discrepancy of the entries (which is a measurement between the sum of the actual entries in blocks versus the expected sum). Our main result is to show that these are related in that discrepancy can be bounded by the second largest singular value and vice versa. These matrix results are then used to derive some (edge/alternating walks) discrepancy properties of edge-weighted directed graphs
Expander Graphs and Coding Theory
Expander graphs are highly connected sparse graphs which lie at the interface of many different fields of study. For example, they play important roles in prime sieves, cryptography, compressive sensing, metric embedding, and coding theory to name a few. This thesis focuses on the connections between sparse graphs and coding theory. It is a major challenge to explicitly construct sparse graphs with good expansion properties, for example Ramanujan graphs. Nevertheless, explicit constructions do exist, and in this thesis, we survey many of these constructions up to this point including a new construction which slightly improves on an earlier edge expansion bound. The edge expansion of a graph is crucial in applications, and it is well-known that computing the edge expansion of an arbitrary graph is NP-hard. We present a simple algo-rithm for approximating the edge expansion of a graph using linear programming techniques. While Andersen and Lang (2008) proved similar results, our analysis attacks the problem from a different vantage point and was discovered independently. The main contribution in the thesis is a new result in fast decoding for expander codes. Current algorithms in the literature can decode a constant fraction of errors in linear time but require that the underlying graphs have vertex expansion at least 1/2. We present a fast decoding algorithm that can decode a constant fraction of errors in linear time given any vertex expansion (even if it is much smaller than 1/2) by using a stronger local code, and the fraction of errors corrected almost doubles that of Viderman (2013)
Spectrum Preserving Short Cycle Removal on Regular Graphs
We describe a new method to remove short cycles on regular graphs while maintaining spectral bounds (the nontrivial eigenvalues of the adjacency matrix), as long as the graphs have certain combinatorial properties. These combinatorial properties are related to the number and distance between short cycles and are known to happen with high probability in uniformly random regular graphs.
Using this method we can show two results involving high girth spectral expander graphs. First, we show that given d ? 3 and n, there exists an explicit distribution of d-regular ?(n)-vertex graphs where with high probability its samples have girth ?(log_{d-1} n) and are ?-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by 2?{d-1} + ? (excluding the single trivial eigenvalue of d). Then, for every constant d ? 3 and ? > 0, we give a deterministic poly(n)-time algorithm that outputs a d-regular graph on ?(n)-vertices that is ?-near-Ramanujan and has girth ?(?{log n}), based on the work of [Mohanty et al., 2020]
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