Understanding and interpreting classification decisions of automated image classification systems is of high value in many applications, as it allows to verify the reasoning of the system and provides additional information to the human expert. Although machine learning methods are solving very successfully a plethora of tasks, they have in most cases the disadvantage of acting as a black box, not providing any information about what made them arrive at a particular decision. This work proposes a general solution to the problem of understanding classification decisions by pixel-wise decomposition of nonlinear classifiers. We introduce a methodology that allows to visualize the contributions of single pixels to predictions for kernel-based classifiers over Bag of Words features and for multilayered neural networks. These pixel contributions can be visualized as heatmaps and are provided to a human expert who can intuitively not only verify the validity of the classification decision, but also focus further analysis on regions of potential interest. We evaluate our method for classifiers trained on PASCAL VOC 2009 images, synthetic image data containing geometric shapes, the MNIST handwritten digits data set and for the pre-trained ImageNet model available as part of the Caffe open source package.

We introduce a machine learning model to predict atomization energies of a diverse set of organic molecules, based on nuclear charges and atomic positions only. The problem of solving the molecular Schrödinger equation is mapped onto a non-linear statistical regression problem of reduced complexity. Regression models are trained on and compared to atomization energies computed with hybrid density-functional theory. Cross-validation over more than seven thousand small organic molecules yields a mean absolute error of ∼10 kcal/mol. Applicability is demonstrated for the prediction of molecular atomization potential energy curves.Solving the Schrödinger equation (SE), HΨ = EΨ, for assemblies of atoms is a fundamental problem in quantum mechanics. Alas, solutions that are exact up to numerical precision are intractable for all but the smallest systems with very few atoms. Hierarchies of approximations have evolved, usually trading accuracy for computational efficiency [1]. Conventionally, the external potential, defined by a set of nuclear charges {Z I } and atomic positions {R I }, uniquely determines the Hamiltonian H of any system, and thereby the potential energy by optimizingFor a diverse set of organic molecules, we show that one can use machine learning (ML) instead, {Z I , R I } ML −→ E. Thus, we circumvent the task of explicitly solving the SE by training once a machine on a finite subset of known solutions. Since many interesting questions in physics require to repeatedly solve the SE, the highly competitive performance of our ML approach may pave the way to large scale exploration of molecular energies in chemical compound space [3,4]. ML techniques have recently been used with success to map the problem of solving complex physical differential equations to statistical models. Successful attempts include solving Fokker-Planck stochastic differential equations [5], parameterizing interatomic force fields for fixed chemical composition [6,7], and the discovery of novel ternary oxides for batteries [8]. Motivated by these, and other related efforts [9-12], we develop a non-linear regression ML model for computing molecular atomization energies in chemical compound space [3]. Our model is based on a measure of distance in compound space that accounts for both stoichiometry and configurational variation. After training, energies are predicted for new (out-of-sample) molecular systems, differing in composition and geometry, at negligible computational cost, i.e. milli seconds instead of hours on a conventional CPU. While the model is trained and tested using atomization energies calculated at the hybrid density-functional theory (DFT) level [2,13,14], any other training set or level of theory could be used as a starting point for subsequent ML training. Cross-validation on 7165 molecules yields a mean absolute error of 9.9 kcal/mol, which is an order of magnitude more accurate than counting bonds or semi-empirical quantum chemistry.We use the GDB data base, a library of nearly one billion organic molecules that ...