# Optimal randomizations in quantizer design with marginal constraint

**Naci Saldi**, Queen's University, Canada

**Abstract:**

We investigate the problem of optimal randomized quantization with a constraint on the marginal distribution of the output. This problem can be cast as a constrained optimal mass transportation problem. First, quantizers are characterized by their induced probability measures on the product of the input and out spaces. This set is proved to be a measurable subset of the set of all probability measures on the product space, which enables us to endow it with a probability measure, giving rise to a general model for randomized quantization. We then parameterize this set over the unit interval, which essentially corresponds to nonuniform dithering. Using results from Optimal Transport Theory, we show the existence of an optimal randomized quantization policy with a fixed output marginal when the source distribution is atomless, the distortion function is non-negative and continuous, and the output space is compact. We then relax the fixed output marginal condition and consider the problem where the output marginal belongs to some neighborhood (in the weak topology) of a fixed probability measure on the output space. We investigate the structure of optimal randomized quantization in this case and show that finitely randomized quantizers form an optimal class for this relaxed minimization problem. This is joint work with Tamas Linder and Serdar Yüksel.

#### Presentation Slides

**Biography:**

Naci Saldi was born in March 23, 1985 in Zonguldak, Turkey. He graduated from the Electrical

and Electronics Engineering Department at Bilkent University, Ankara, in 2008. He got his M.S.

degree in the same field again at Bilkent University, in 2010 under the supervision of Prof. Morgul. He is currently a second year Ph.D. student in Mathematics and Statistics Department at Queen’s University under the supervision of Prof. Yüksel and Linder.