The first work is a type of "derivation from the Fourier Transform problem". It is documented in a manuscript:
“The Complexity of Computing a Fourier Perturbation” (Together with Gal Yehuda). This paper studies the problem of computing a perturbation of the Fourier transform, defined as the identity matrix plus epsilon (a small perturbation number) plus the Fourier transform matrix. Previous work by the PI showed that a lower bound of epsilon^2 n log n exists for computing the perturbation transformation, using the quasi-entropy method (invented by the PI). However, the result in the paper implies the stronger lower bound of n log n / log(1/epsilon). This seems optimal, because computing with accuracy epsilon requires log(1/epsilon) bits, which cancels out the log(1/epsilon) speedup in the running time. Unfortunately the paper’s result works only for the case that F is a symmetric transform Fourier. For the case that F is asymmetric, the result does not generalize. An important open problem which the PI is working on now is how to generalize the result to the asymmetric case.
In the second work, the PI looked closely at the notion of "condition number" that is often assumed in literature, which is a measure of numerical accuracy of computation. He identifies a more reasonable notion, called "algebraic condition number" that lends itself to improved lower bounds. This is described int the manuscript: “Paraunitary Matrices, Entropy, Algebraic Condition Number and Fourier Computation”: This work has appeared in Journal TCS (Theoretical Computer Science) pending minor revisions, as of June 2019.
Additionally, the following results have been obtained:
- In work under submission, it a tradeoff has been show between dimensionality reduction and NTK (Neural Tangent Kernel) guarantees.
- Various papers on the subject of information theory in the context of graph properties have been accepted to conferences UAI, FCT, WALCOM and AAAI
- Various papers on the subject of information theory in communication have been accepted to ISIT conference.
- A paper in the subject of information tradeoff between accuracy and speed of deep networks has been accepted to conference UAI.
- Various papers in the topic of tradeoffs between side information and the computational ability to take advantage of such information in the context of cooperating agents has been accepted to conferences AAMAS, AISTATS and ICML.
