Duistermaat-Heckman measure and the mixture of quantum states

In this paper, we present a general framework to solve a fundamental problem in Random Matrix Theory (RMT), i.e., the problem of describing the joint distribution of eigenvalues of the sum \bsA+\bsB of two independent random Hermitian matrices \bsA and \bsB. Some considerations about the mixture of quantum states are basically subsumed into the above mathematical problem. Instead, we focus on deriving the spectral density of the mixture of adjoint orbits of quantum states in terms of Duistermaat-Heckman measure, originated from the theory of symplectic geometry. Based on this method, we can obtain the spectral density of the mixture of independent random states. In particular, we obtain explicit formulas for the mixture of random qubits. We also find that, in the two-level quantum system, the average entropy of the equiprobable mixture of $n$ random density matrices chosen from a random state ensemble (specified in the text) increases with the number $n$. Hence, as a physical application, our results quantitatively explain that the quantum coherence of the mixture monotonously decreases statistically as the number of components $n$ in the mixture. Besides, our method may be used to investigate some statistical properties of a special subclass of unital qubit channels.Comment: 40 pages, 10 figures, LaTeX, the final version accepted for publication in J. Phys.

Rethinking Medical Report Generation: Disease Revealing Enhancement with Knowledge Graph

Knowledge Graph (KG) plays a crucial role in Medical Report Generation (MRG) because it reveals the relations among diseases and thus can be utilized to guide the generation process. However, constructing a comprehensive KG is labor-intensive and its applications on the MRG process are under-explored. In this study, we establish a complete KG on chest X-ray imaging that includes 137 types of diseases and abnormalities. Based on this KG, we find that the current MRG data sets exhibit a long-tailed problem in disease distribution. To mitigate this problem, we introduce a novel augmentation strategy that enhances the representation of disease types in the tail-end of the distribution. We further design a two-stage MRG approach, where a classifier is first trained to detect whether the input images exhibit any abnormalities. The classified images are then independently fed into two transformer-based generators, namely, ``disease-specific generator" and ``disease-free generator" to generate the corresponding reports. To enhance the clinical evaluation of whether the generated reports correctly describe the diseases appearing in the input image, we propose diverse sensitivity (DS), a new metric that checks whether generated diseases match ground truth and measures the diversity of all generated diseases. Results show that the proposed two-stage generation framework and augmentation strategies improve DS by a considerable margin, indicating a notable reduction in the long-tailed problem associated with under-represented diseases

Modified Kedem-Katchalsky equations for osmosis through nano-pore

This work presents a modified Kedem-Katchalsky equations for osmosis through nano-pore. osmotic reflection coefficient of a solute was found to be chiefly affected by the entrance of the pore while filtration reflection coefficient can be affected by both the entrance and the internal structure of the pore. Using an analytical method, we get the quantitative relationship between osmotic reflection coefficient and the molecule size. The model is verified by comparing the theoretical results with the reported experimental data of aquaporin osmosis. Our work is expected to pave the way for a better understanding of osmosis in bio-system and to give us new ideas in designing new membranes with better performance.Comment: 19 pages, 4 figure

Business Policy Experiments using Fractional Factorial Designs: Consumer Retention on DoorDash

This paper investigates an approach to both speed up business decision-making and lower the cost of learning through experimentation by factorizing business policies and employing fractional factorial experimental designs for their evaluation. We illustrate how this method integrates with advances in the estimation of heterogeneous treatment effects, elaborating on its advantages and foundational assumptions. We empirically demonstrate the implementation and benefits of our approach and assess its validity in evaluating consumer promotion policies at DoorDash, which is one of the largest delivery platforms in the US. Our approach discovers a policy with 5% incremental profit at 67% lower implementation cost.Comment: 14 page

Online Bayesian Analysis

In the last few years, there has been active research on aggregating advanced statistical measures in multidimensional data cubes from partitioned subsets of data. In this paper, we propose an online compression and aggregation scheme to support Bayesian estimations in data cubes based on the asymptotic properties of Bayesian statistics. In the proposed approach, we compress each data segment by retaining only the model parameters and a small amount of auxiliary measures. We then develop an aggregation formula that allows us to reconstruct the Bayesian estimation from partitioned segments with a small approximation error. We show that the Bayesian estimates and the aggregated Bayesian estimates are asymptotically equivalent