arXiv subjects

Michael Mara

Publications and source records attributed to Michael Mara.

3 recordsLinked to original sources

arXiv2020

Differentiating a Tensor Language

How does one compile derivatives of tensor programs, such that the resulting code is purely functional (hence easier to optimize and parallelize) and provably efficient relative to the original program? We show that naively differentiating tensor code---as done in popular systems like Tensorflow and PyTorch---can cause asymptotic slowdowns in pathological cases, violating the Cheap Gradients Principle. However, all existing automatic differentiation methods that guarantee this principle (for variable size data) do so by relying on += mutation through aliases/pointers---which complicates downstream optimization. We provide the first purely functional, provably efficient, adjoint/reverse-mode derivatives of array/tensor code by explicitly accounting for sparsity. We do this by focusing on the indicator function from Iverson's APL. We also introduce a new "Tensor SSA" normal form and a new derivation of reverse-mode automatic differentiation based on the universal property of inner-products.

cs.PL↗

arXiv2017

Opt: A Domain Specific Language for Non-linear Least Squares Optimization in Graphics and Imaging

Many graphics and vision problems can be expressed as non-linear least squares optimizations of objective functions over visual data, such as images and meshes. The mathematical descriptions of these functions are extremely concise, but their implementation in real code is tedious, especially when optimized for real-time performance on modern GPUs in interactive applications. In this work, we propose a new language, Opt (available under http://optlang.org), for writing these objective functions over image- or graph-structured unknowns concisely and at a high level. Our compiler automatically transforms these specifications into state-of-the-art GPU solvers based on Gauss-Newton or Levenberg-Marquardt methods. Opt can generate different variations of the solver, so users can easily explore tradeoffs in numerical precision, matrix-free methods, and solver approaches. In our results, we implement a variety of real-world graphics and vision applications. Their energy functions are expressible in tens of lines of code, and produce highly-optimized GPU solver implementations. These solver have performance competitive with the best published hand-tuned, application-specific GPU solvers, and orders of magnitude beyond a general-purpose auto-generated solver.

cs.GR↗

arXiv2011

Isoperimetric Pentagonal Tilings

We identify least-perimeter unit-area tilings of the plane by convex pentagons, namely tilings by Cairo and Prismatic pentagons, find infinitely many, and prove that they minimize perimeter among tilings by convex polygons with at most five sides.

math.MG↗