6,121 research outputs found
Ramanujan Graphs in Polynomial Time
The recent work by Marcus, Spielman and Srivastava proves the existence of
bipartite Ramanujan (multi)graphs of all degrees and all sizes. However, that
paper did not provide a polynomial time algorithm to actually compute such
graphs. Here, we provide a polynomial time algorithm to compute certain
expected characteristic polynomials related to this construction. This leads to
a deterministic polynomial time algorithm to compute bipartite Ramanujan
(multi)graphs of all degrees and all sizes
Online Row Sampling
Finding a small spectral approximation for a tall matrix is
a fundamental numerical primitive. For a number of reasons, one often seeks an
approximation whose rows are sampled from those of . Row sampling improves
interpretability, saves space when is sparse, and preserves row structure,
which is especially important, for example, when represents a graph.
However, correctly sampling rows from can be costly when the matrix is
large and cannot be stored and processed in memory. Hence, a number of recent
publications focus on row sampling in the streaming setting, using little more
space than what is required to store the outputted approximation [KL13,
KLM+14].
Inspired by a growing body of work on online algorithms for machine learning
and data analysis, we extend this work to a more restrictive online setting: we
read rows of one by one and immediately decide whether each row should be
kept in the spectral approximation or discarded, without ever retracting these
decisions. We present an extremely simple algorithm that approximates up to
multiplicative error and additive error using online samples, with memory overhead
proportional to the cost of storing the spectral approximation. We also present
an algorithm that uses ) memory but only requires
samples, which we show is
optimal.
Our methods are clean and intuitive, allow for lower memory usage than prior
work, and expose new theoretical properties of leverage score based matrix
approximation
Optimal approximate matrix product in terms of stable rank
We prove, using the subspace embedding guarantee in a black box way, that one
can achieve the spectral norm guarantee for approximate matrix multiplication
with a dimensionality-reducing map having
rows. Here is the maximum stable rank, i.e. squared ratio of
Frobenius and operator norms, of the two matrices being multiplied. This is a
quantitative improvement over previous work of [MZ11, KVZ14], and is also
optimal for any oblivious dimensionality-reducing map. Furthermore, due to the
black box reliance on the subspace embedding property in our proofs, our
theorem can be applied to a much more general class of sketching matrices than
what was known before, in addition to achieving better bounds. For example, one
can apply our theorem to efficient subspace embeddings such as the Subsampled
Randomized Hadamard Transform or sparse subspace embeddings, or even with
subspace embedding constructions that may be developed in the future.
Our main theorem, via connections with spectral error matrix multiplication
shown in prior work, implies quantitative improvements for approximate least
squares regression and low rank approximation. Our main result has also already
been applied to improve dimensionality reduction guarantees for -means
clustering [CEMMP14], and implies new results for nonparametric regression
[YPW15].
We also separately point out that the proof of the "BSS" deterministic
row-sampling result of [BSS12] can be modified to show that for any matrices
of stable rank at most , one can achieve the spectral norm
guarantee for approximate matrix multiplication of by deterministically
sampling rows that can be found in polynomial
time. The original result of [BSS12] was for rank instead of stable rank. Our
observation leads to a stronger version of a main theorem of [KMST10].Comment: v3: minor edits; v2: fixed one step in proof of Theorem 9 which was
wrong by a constant factor (see the new Lemma 5 and its use; final theorem
unaffected
Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods
In this paper, we study matrix scaling and balancing, which are fundamental
problems in scientific computing, with a long line of work on them that dates
back to the 1960s. We provide algorithms for both these problems that, ignoring
logarithmic factors involving the dimension of the input matrix and the size of
its entries, both run in time where is the amount of error we are willing to
tolerate. Here, represents the ratio between the largest and the
smallest entries of the optimal scalings. This implies that our algorithms run
in nearly-linear time whenever is quasi-polynomial, which includes, in
particular, the case of strictly positive matrices. We complement our results
by providing a separate algorithm that uses an interior-point method and runs
in time .
In order to establish these results, we develop a new second-order
optimization framework that enables us to treat both problems in a unified and
principled manner. This framework identifies a certain generalization of linear
system solving that we can use to efficiently minimize a broad class of
functions, which we call second-order robust. We then show that in the context
of the specific functions capturing matrix scaling and balancing, we can
leverage and generalize the work on Laplacian system solving to make the
algorithms obtained via this framework very efficient.Comment: To appear in FOCS 201
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