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Eugene Katsevich works on high-dimensional statistics, variable selection and multiple testing, motivated in part by genetics and genomics. These problems require distinguishing reproducible signals from the many associations that can arise when thousands of features are examined.^[ref]

His joint work on the conditional randomization test uses a known feature conditional distribution to calibrate tests of conditional independence. Distillation makes flexible prediction methods more practical as test statistics.^[ref]

For response vector $Y$, design matrix $X$ and $\lambda \ge 0$, the Lasso estimates regression coefficients byAn equivalent constrained formulation minimizes the squared residual norm subject to $\Vert \beta {\Vert }_1\le t$, with an appropriate matching constraint radius. The L1 norm has a subdifferential interval at zero, allowing exact zero regression coefficients rather than only small ones. This combines estimation with variable selection.^[ref]

For an orthonormalized design matrix, $X^{\mathsf{T}}X/n=I$, put $z=X^{\mathsf{T}}Y/n$. Expanding the objective separates its coordinates and givesthe soft-thresholding operator. Thus $z=(2,0.4)$ and $\lambda =0.5$ give $\hat{\beta }=(1.5,0)$. With correlated columns, the Karush-Kuhn-Tucker conditions for the Lasso replace this independent-coordinate calculation. An unpenalized regression intercept is ordinarily handled by centering.

An intercept-unpenalized Lasso estimator solves the constrained optimizationFor centered predictors, eliminate the intercept as in ridge regression and minimize $\Vert Y_c-X\beta {\Vert }_2^2$ under the same constraint. Equivalently, with a suitable tuning parameter $\lambda \ge 0$, use $\Vert Y-\alpha 1-X\beta {\Vert }_2^2+\lambda {\sum }_j|{\beta }_j|$. The constraint radius and penalty parameter are different parametrizations; larger radius permits less shrinkage.

The Lasso plot parametrizes the Lasso regularization path by the fraction of the maximum $l_1$ norm of the standardized slopes. Unlike the ridge quadratic penalty, the corners of the $l_1$ constraint can place some slopes exactly at zero, providing variable selection. The maximum norm refers to the path's unpenalized endpoint; centering and scaling conventions must match those used to construct that path.

Sparse optimization seeks solutions with few nonzero coordinates, either as an explicit objective or through a penalty. Minimizing the number of nonzero entries, written $\Vert x{\Vert }_0$, is combinatorial; replacing it by the L1 norm often gives a tractable convex optimization problem.

For example, basis pursuit minimizes $\Vert x{\Vert }_1$ subject to $Ax=b$, while Lasso minimizes a squared residual plus $\lambda \Vert x{\Vert }_1$. The penalty favors exact zero coordinates because it has a corner at zero. Recovery of the sparsest explanation needs assumptions on $A$; an L1 norm objective alone does not establish it. These methods connect variable selection with compressed sensing.