General, Nested, and Constrained Wiberg Minimization

Wiberg matrix factorization breaks a matrix <inline-formula><tex-math notation="LaTeX">$Y$</tex-math> <alternatives><inline-graphic xlink:type="simple" xlink:href="strelow-ieq1-2487987.gif"/></alternatives></inline-formula> into low-rank factors <inline-formula><tex-math notation="LaTeX">$U$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="strelow-ieq2-2487987.gif"/></alternatives></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$V$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="strelow-ieq3-2487987.gif"/></alternatives></inline-formula> by solving for <inline-formula> <tex-math notation="LaTeX">$V$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="strelow-ieq4-2487987.gif"/></alternatives></inline-formula> in closed form given <inline-formula><tex-math notation="LaTeX">$U$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="strelow-ieq5-2487987.gif"/></alternatives></inline-formula>, linearizing <inline-formula> <tex-math notation="LaTeX">$V(U)$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="strelow-ieq6-2487987.gif"/></alternatives></inline-formula> about <inline-formula> <tex-math notation="LaTeX">$U$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="strelow-ieq7-2487987.gif"/></alternatives></inline-formula>, and iteratively minimizing <inline-formula><tex-math notation="LaTeX">$||Y - UV(U)||_2$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="strelow-ieq8-2487987.gif"/></alternatives></inline-formula> with respect to <inline-formula> <tex-math notation="LaTeX">$U$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="strelow-ieq9-2487987.gif"/></alternatives></inline-formula> only. This approach factors the matrix while effectively removing <inline-formula><tex-math notation="LaTeX">$V$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="strelow-ieq10-2487987.gif"/></alternatives></inline-formula> from the minimization. Recently Eriksson and van den Hengel extended this approach to <inline-formula><tex-math notation="LaTeX"> $L_1$</tex-math><alternatives><inline-graphic xlink:type="simple" xlink:href="strelow-ieq11-2487987.gif"/></alternatives> </inline-formula>, minimizing <inline-formula><tex-math notation="LaTeX">$||Y - UV(U)||_1$</tex-math> <alternatives><inline-graphic xlink:type="simple" xlink:href="strelow-ieq12-2487987.gif"/></alternatives></inline-formula>. We generalize their approach beyond factorization to minimize <inline-formula><tex-math notation="LaTeX"> $||Y - f(U, V)||_1$</tex-math><alternatives><inline-graphic xlink:type="simple" xlink:href="strelow-ieq13-2487987.gif"/> </alternatives></inline-formula> for more general functions <inline-formula><tex-math notation="LaTeX"> $f(U, V)$</tex-math><alternatives><inline-graphic xlink:type="simple" xlink:href="strelow-ieq14-2487987.gif"/></alternatives> </inline-formula> that are nonlinear in each of two sets of variables. We demonstrate the idea with a practical Wiberg algorithm for <inline-formula><tex-math notation="LaTeX">$L_1$</tex-math><alternatives> <inline-graphic xlink:type="simple" xlink:href="strelow-ieq15-2487987.gif"/></alternatives></inline-formula> bundle adjustment. One Wiberg minimization can be nested inside another, effectively removing two of three sets of variables from a minimization. We demonstrate this idea with a nested Wiberg algorithm for <inline-formula><tex-math notation="LaTeX">$L_1$ </tex-math><alternatives><inline-graphic xlink:type="simple" xlink:href="strelow-ieq16-2487987.gif"/></alternatives></inline-formula> projective bundle adjustment, solving for camera matrices, points, and projective depths. Wiberg minimization also generalizes to handle nonlinear constraints, and we demonstrate this idea with Constrained Wiberg Minimization for Multiple Instance Learning (CWM-MIL), which removes one set of variables from the constrained optimization. Our experiments emphasize isolating the effect of Wiberg by comparing against the algorithm it modifies, successive linear programming.