Dimensionless machine learning: Imposing exact units equivariance

Units equivariance is the exact symmetry that follows from the requirement that relationships among measured quantities of physics relevance must obey self-consistent dimensional scalings. Here, we employ dimensional analysis and ideas from equivariant machine learning to provide a two stage learning procedure for units-equivariant machine learning. For a given learning task, we first construct a dimensionless version of its inputs using classic results from dimensional analysis, and then perform inference in the dimensionless space. Our approach can be used to impose units equivariance across a broad range of machine learning methods which are equivariant to rotations and other groups. We discuss the in-sample and out-of-sample prediction accuracy gains one can obtain in contexts like symbolic regression and emulation, where symmetry is important. We illustrate our approach with simple numerical examples involving dynamical systems in physics and ecology

Recurrent melanocytic nevi and melanomas in dermoscopy: results of a multicenter study of the International Dermoscopy Society

IMPORTANCE Differentiating recurrent nevi from recurrent melanoma is challenging. OBJECTIVE To determine dermoscopic features to differentiate recurrent nevi from melanomas. DESIGN, SETTING, AND PARTICIPANTS Retrospective observational study of 15 pigmented lesion clinics from 12 countries; 98 recurrent nevi (61.3%) and 62 recurrent melanomas (38.8%) were collected from January to December 2011. MAIN OUTCOMES AND MEASURES Scoring the dermoscopic features, patterns, and colors in correlation with the histopathologic findings. RESULTS In univariate analysis, radial lines, symmetry, and centrifugal growth pattern were significantly more common dermoscopically in recurrent nevi; in contrast, circles, especially if on the head and neck area, eccentric hyperpigmentation at the periphery, a chaotic and noncontinuous growth pattern, and pigmentation beyond the scar's edge were significantly more common in recurrent melanomas. Patients with recurrent melanomas were significantly older than patients with recurrent nevi (mean [SD] age, 63.1 [17.5] years vs 30.2 [12.4] years) (P<.001), and there was a significantly longer time interval between the first procedure and the second treatment (median time interval, 25 vs 8 months) (P<.001). In a multivariate analysis, pigmentation beyond the scar's edge (P=.002), age (P<.001), and anatomic site (P=.002) were significantly and independently associated with the diagnosis of recurrent melanoma in dermoscopy. CONCLUSIONS AND RELEVANCE Dermoscopically, pigmentation beyond the scar's edge is the strongest clue for melanoma. Dermoscopy is helpful in evaluating recurrent lesions, but final interpretation requires taking into account the patient age, anatomic site, time to recurrence, growth pattern, and, if available, the histopathologic findings of the first excision