Hand-printed digits can be modeled as splines that are governed by about 8 control points. Images of digits can be produced by placing Gaussian ink generators uniformly along the spline. Real images can be recognized by finding the digit model most likely to have generated the data. For each digit model we use an elastic matching algorithm to minimize an energy function that includes both the defor(cid:173) mation energy of the digit model and the log probability that the model would generate the inked pixels in the image. If a uniform noise process is included in the model of image generation, some of the inked pixels can be rejected as noise as a digit model is fitting a poorly segmented image.

Symbolic knowledge can provide crucial inductive bias for training neural models, especially in low data regimes. A successful strategy for incorporating such knowledge involves relaxing logical statements into sub-differentiable losses for optimization. In this paper, we study the question of how best to relax logical expressions that represent labeled examples and knowledge about a problem; we focus on sub-differentiable t-norm relaxations of logic. We present theoretical and empirical criteria for characterizing which relaxation would perform best in various scenarios. In our theoretical study driven by the goal of preserving tautologies, the Lukasiewicz t-norm performs best. However, in our empirical analysis on the text chunking and digit recognition tasks, the product t-norm achieves best predictive performance. We analyze this apparent discrepancy, and conclude with a list of best practices for defining loss functions via logic.

Hand-printed digits can be modeled as splines that are governed by about 8 control points.

Hand-printed digits can be modeled as splines that are governed by about 8 control points.

Hand-printed digits can be modeled as splines that are governed by about 8 control points.