Selected Code

Discrete Image Coding Model

(with Ram Mehta and Kilian Koepsell) A Hopfield recurrent neural network trained on natural images performs state-of-the-art image compression, IEEE International Conference on Image Processing (ICIP), 2014, pp. 4092-4096.

Python code implementing mean SSIM used in above paper: mssim.py

Efficient / Exponential Hopfield network learning

You'll want to get Python, Numpy, etc, all conveniently contained in this distribution Anaconda (free).

Hopfield network Python class with (exponential capacity) fitting using minimum probability flow learning: local_mpf_rule.txt

(with Ngoc Tran) Robust exponential memory in Hopfield networks, 2014, submitted. pdf |arxiv

(with Jascha Sohl-Dickstein and Kilian Koepsell) Efficient and optimal binary Hopfield associative memory storage using minimum probability flow, 2011, NIPS (DISCML Workshop), 2012. pdf | arxiv.

Most tensor problems are NP-hard

The following code verifies Example 1.5 and Lemma 7.1 in the following paper:

(with L.H. Lim) Most tensor problems are NP-hard, Journal of the ACM, 60 (2013), no. 6, Art. 45, 39 pp. pdf | Ex 1.5 SINGULAR code | Appendix SINGULAR code, Macaulay 2 code

Finiteness theorems and algorithms for permutation invariant chains

The following Macaulay 2 code computes the table found in the paper:

(with A. Martin del Campo) Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals, Journal of Symbolic Computation, 50 (2013) 314-334. pdf | errata (soon!)

Macaulay 2 code | Table Generation code

Solvability of symmetric word equations

The following Maple code computes Jacobians (and subspace restrictions of Jacobians) for Words in matrix letters. You might need to "right-click save as" download these files (as your browser might think they are readable).

maple code 1 | maple code 2

The first code listed above verified a calculation showing that there are word equations in positive definite letters with multiple postiive definite solutions. This settled an open conjecture. The second piece of code gives evidence for the conjecture that in the 2-by-2 case, there is always a unique solution. These results can be found in the paper:

(with S. Armstrong). Solvability of symmetric word equations in positive definite letters, Journal of the London Mathematical Society, 76 (2007), no. 3, 777-796. arXiv | pdf

Algebraic Characterization of Uniquely Colorable Graphs

The following Singular code verifies a counterexample to a conjecture of Xu discovered by Akbari, Mirrokni, and Sadjad.

singular code

It uses Groebner basis techniques to discover unique colorability of graphs. The details can be found in the following paper:

(with T. Windfeldt). An algebraic characterization of uniquely vertex colorable graphs, Journal of Combinatorial Theory Series B, 98 (2008), 400-414. pdf | arXiv

Introduction to Maple

A basic introduction to Maple (there is some code here). pdf