Please note: This seminar will take place online.
Iden Kalemaj, PhD candidate
Department of Computer Science, Boston University
We study sublinear algorithms for monotonicity of real-valued functions. An algorithm for testing monotonicity queries a function at a few locations and distinguishes between when the function is monotone and when it is far from monotone. An algorithm for approximating the distance to monotonicity returns the distance to monotonicity of the function with some additive and multiplicative error. We improve previous algorithms for these two tasks by showing that several isoperimetric inequalities about Boolean functions on the hypercube can be generalized to real-valued functions.
A celebrated line of work [Margulis ’74, Talagrand ’93, Chakrabarty and Seshadhri ’13, Khot Minzer Safra ’15] studied the size of the “boundary'” between the points on which a Boolean function takes value 0 and the points on which it takes value 1. This boundary is defined in terms of the edges of the d-dimensional hypercube, where the edges can be directed or undirected. In particular, the inequality of Khot, Minzer, and Safra ’15 implies all previous inequalities. It bounds the average square root of the number of decreasing edges incident on a vertex of the hypercube by the distance to monotonicity of the function. We generalize all the inequalities in this line of work to real-valued functions. Our main contribution is a Boolean decomposition technique that represents a real-valued function as a collection of Boolean functions that roughly capture the distance to monotonicity of the original function. In this talk, we prove the Boolean decomposition theorem and touch on the improved algorithms for monotonicity testing and distance approximation.
This is joint work with Hadley Black and Sofya Raskhodnikova.
To join this seminar on Zoom, please go to https://bostonu.zoom.us/j/93466741386.
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