Three-dimensional Radial Visualization of High-dimensional Datasets with Mixed Features

We develop methodology for 3D radial visualization (RadViz) of high-dimensional datasets. Our display engine is called RadViz3D and extends the classical 2D RadViz that visualizes multivariate data in the 2D plane by mapping every record to a point inside the unit circle. We show that distributing anchor points at least approximately uniformly on the 3D unit sphere provides a better visualization with minimal artificial visual correlation for data with uncorrelated variables. Our RadViz3D methodology therefore places equi-spaced anchor points, one for every feature, exactly for the five Platonic solids, and approximately via a Fibonacci grid for the other cases. Our Max-Ratio Projection (MRP) method then utilizes the group information in high dimensions to provide distinctive lower-dimensional projections that are then displayed using Radviz3D. Our methodology is extended to datasets with discrete and continuous features where a Gaussianized distributional transform is used in conjunction with copula models before applying MRP and visualizing the result using RadViz3D. A R package radviz3d implementing our complete methodology is available.Comment: 12 pages, 10 figures, 1 tabl

Communication Protocols under Transparent Motives

We study optimal (information) mediation in sender-receiver communication games where the sender has transparent motives: she only cares about the receiver's actions and beliefs. We characterize the feasible distributions over the receiver's beliefs under mediation and the value of mediation. The sender achieves her optimal Bayesian persuasion value by mediation if and only if this value is attained by cheap talk. When the state is binary, mediation strictly improves on cheap talk if and only if the sender cannot do better than under cheap talk by always under -- or over -- stating the state

Unconstrained Proximal Operator: the Optimal Parameter for the Douglas-Rachford Type Primal-Dual Methods

In this work, we propose an alternative parametrized form of the proximal operator, of which the parameter no longer needs to be positive. That is, the parameter can be a non-zero scalar, a full-rank square matrix, or, more generally, a bijective bounded linear operator. We demonstrate that the positivity requirement is essentially due to a quadratic form. We prove several key characterizations for the new form in a generic way (with an operator parameter). We establish the optimal choice of parameter for the Douglas-Rachford type methods by solving a simple unconstrained optimization problem. The optimality is in the sense that a non-ergodic worst-case convergence rate bound is minimized. We provide closed-form optimal choices for scalar and orthogonal matrix parameters under zero initialization. Additionally, a simple self-contained proof of a sharp linear convergence rate for a $(1/L)$-cocoercive fixed-point sequence with $L\in(0,1)$ is provided (as a preliminary result). To our knowledge, an operator parameter is new. To show its practical use, we design a dedicated parameter for the 2-by-2 block-structured semidefinite program (SDP). Such a structured SDP is strongly related to the quadratically constrained quadratic program (QCQP), and we therefore expect the proposed parameter to be of great potential use. At last, two well-known applications are investigated. Numerical results show that the theoretical optimal parameters are close to the practical optimums, except they are not a priori knowledge. We then demonstrate that, by exploiting problem model structures, the theoretical optimums can be well approximated. Such approximations turn out to work very well, and in some cases almost reach the underlying limits

Three-dimensional Radial Visualization of High-dimensional Continuous or Discrete Data

This paper develops methodology for 3D radial visualization of high-dimensional datasets. Our display engine is called RadViz3D and extends the classic RadViz that visualizes multivariate data in the 2D plane by mapping every record to a point inside the unit circle. The classic RadViz display has equally-spaced anchor points on the unit circle, with each of them associated with an attribute or feature of the dataset. RadViz3D obtains equi-spaced anchor points exactly for the five Platonic solids and approximately for the other cases via a Fibonacci grid. We show that distributing anchor points at least approximately uniformly on the 3D unit sphere provides a better visualization than in 2D. We also propose a Max-Ratio Projection (MRP) method that utilizes the group information in high dimensions to provide distinctive lower-dimensional projections that are then displayed using Radviz3D. Our methodology is extended to datasets with discrete and mixed features where a generalized distributional transform is used in conjuction with copula models before applying MRP and RadViz3D visualization