Ethan Epperly

Miller research fellow at UC Berkeley. Interested in computational linear algebra he/him

[Ethan Epperly @ethanepperly]

Randomized block Krylov is great at low-rank approximation. But existing analysis gives good results for only large and small block sizes, even though the best block size in practice tends to be in the middle. We resolve this theory-practice gap in https://t.co/iYTshDEGXv

[Ethan Epperly @ethanepperly]

@TaliaRinger In (randomized) linear algebra, we have lots of examples of nice algorithms and information-theoretic lower bounds with small constants. Sometimes, even, we can get the constants to (nearly) match, say up to a factor of five https://t.co/Y0bt72sia3

[Ethan Epperly @ethanepperly]

For anyone who wants to learn a bit more computational linear algebra, I highly recommend Tyler Chen’s book on the Lanczos method. Filled to the brim with small delights https://t.co/rZQn7KZ8Lk

[Ethan Epperly @ethanepperly]

More progress by Spielman–Yu which establishes a *1/quasipolynomial lower bound* on the probability of large growth. Taken together, the Urschel/Spielman–Yu results tell an interesting story. The probability of large growth is small but not that small https://t.co/Y7tdyfb9EZ

[Ethan Epperly @ethanepperly]

John Urschel just showed that the growth factor for Gaussian elimination with partial pivoting is ~n^{1/2} with high probability, resolving a long-standing conjecture of Nick Trefethen! https://t.co/FraoYI6yh2

[Ethan Epperly @ethanepperly]

John Urschel just showed that the growth factor for Gaussian elimination with partial pivoting is ~n^{1/2} with high probability, resolving a long-standing conjecture of Nick Trefethen! https://t.co/FraoYI6yh2

[𝖬𝖺𝗁𝖽𝗂 𝖢𝗁. (bluesky:@mahdi.ch) @mahdi_tcs_]

The what now?!
https://t.co/xHsLmQJkff

[Ethan Epperly @ethanepperly]

The analysis is super clean too. Just standard Girard–Hutchinson bounds plus standard approximate matrix multiplication bounds

[Ethan Epperly @ethanepperly]

A new “algorithm from the book” today from Chen–Halikias–Musco–Persson. You want a good estimate of the squared Frobenius norm? Generate some Gaussian matrices and form this expression. It gets relative error ε in 4/ε matrix–vector multiplications https://t.co/arChN1aA9A

[Ethan Epperly @ethanepperly]

The algorithm is called Hutch#. For those in the know, it’s basically Hutch++ but with approximate matrix multiplication instead of low-rank approximation

[Daniel Kressner @_kreda_]

Linear Systems and Eigenvalue Problems: Open Questions from a Simons Workshop https://t.co/Ap2keEQGKr

[Damek @damekdavis]

cool paper @ethanepperly!

[Ethan Epperly @ethanepperly]

@angeris @damekdavis Hm. Not sure you’re on base here. I assume you’re talking about Bubeck §3.5 stuff? We study Ax=b when A is not pd. So matrix-vector products are not gradients of a convex quadratic. A similar result is known for Krylov algorithms, but we bound against arbitrary rand algorithms

[Ethan Epperly @ethanepperly]

New paper out with Chris Camaño, Raphael Meyer, and Joel Tropp re-examining sketching algorithms! Included: subspace injections as an alternative to subspace embeddings, the theory and practice of sparse sketching, tensor sketching, and much more! https://t.co/gq6z1CKsRW

[Ethan Epperly @ethanepperly]

@magoghm Seems correct but also still uses a lot of explicit integrals, which I’m hoping to avoid…

[Ethan Epperly @ethanepperly]

Anyone know an easy way of proving the identity 𝔼[g₁²/(g₁²+a²g₂²)] = 1/(1+a) where g₁, g₂ are iid standard Gaussians and a > 0? Ideally, I want an approach that avoids explicitly integrating over the Gaussian/chi-square/F pdf

[Ethan Epperly @ethanepperly]

@PengLiangzu This is a nice solution. Somehow, still isn’t quite what I’m after. I’m really hoping for a simple direct argument that uses properties of the Gaussian distribution rather than known relations to other distributions

[Ethan Epperly @ethanepperly]

@mathlfs The full version of Gautschi’s bound is attained if all the locations are in a ray on the complex plane. So that shows the exponential scaling is tight. I don’t know of any lower bounds for arbitrary locations

[Ethan Epperly @ethanepperly]

New blog post out! Vandermonde matrices are famously ill-conditioned, but just how bad are they? In this post, I discuss Gautschi’s 1962 bound showing that Vandermonde matrices are merely exponentially ill-conditioned https://t.co/2IZNfNVrw3

[Ethan Epperly @ethanepperly]

Joint work with my awesome collaborators Tyler Chen, Akash Rao, Raphael Meyer, and Chris Musco!

[Ethan Epperly @ethanepperly]

Randomized block Krylov is great at low-rank approximation. But existing analysis gives good results for only large and small block sizes, even though the best block size in practice tends to be in the middle. We resolve this theory-practice gap in https://t.co/iYTshDEGXv

[Ethan Epperly @ethanepperly]

@humphilomath @jwt0625 That’s a depressing thought…