We consider sketched approximate matrix multiplication and ridge regression in the novel setting of localized sketching, where at any given point, only part of the data matrix is available. This corresponds to a block diagonal structure on the sketching matrix. We show that, under mild conditions, block diagonal sketching matrices require only O(sr/ 2 ) and O(sd λ / ) total sample complexity for matrix multiplication and ridge regression, respectively. This matches the state-of-the-art bounds that are obtained using global sketching matrices. The localized nature of sketching considered allows for different parts of the data matrix to be sketched independently and hence is more amenable to computation in distributed and streaming settings and results in a smaller memory and computational footprint.

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The ongoing pandemic has highlighted the importance of reliable and efficient clinical trials in healthcare. Trial sites, where the trials are conducted, are chosen mainly based on feasibility in terms of medical expertise and access to a large group of patients. More recently, the issue of diversity and inclusion in clinical trials is gaining importance. Different patient groups may experience the effects of a medical drug/ treatment differently and hence need to be included in the clinical trials. These groups could be based on ethnicity, co-morbidities, age, or economic factors. Thus, designing a method for trial site selection that accounts for both feasibility and diversity is a crucial and urgent goal. In this paper, we formulate this problem as a ranking problem with fairness constraints. Using principles of fairness in machine learning, we learn a model that maps a clinical trial description to a ranked list of potential trial sites. Unlike existing fairness frameworks, the group membership of each trial site is non-binary: each trial site may have access to patients from multiple groups. We propose fairness criteria based on demographic parity to address such a multi-group membership scenario. We test our method on 480 real-world clinical trials and show that our model results in a list of potential trial sites that provides access to a diverse set of patients while also ensuing a high number of enrolled patients. * This work was conducted while author was an employee of IQVIA Inc. Correspondence email: rakshith.sharma.s[at]gmail.com 2 All clinical trials specify certain inclusion criteria which describe a set of conditions to be met by patients to be enrolled in the trial and exclusion criteria, which describe conditions that determine patients who cannot be enrolled in the study.