Concepts are key building blocks of higher level human understanding. Explainable AI (XAI) methods have shown tremendous progress in recent years, however, local attribution methods do not allow to identify coherent model behavior across samples and therefore miss this essential component. In this work, we study concept-based explanations and put forward a new definition of concepts as lowdimensional subspaces of hidden feature layers. We novelly apply sparse subspace clustering to discover these concept subspaces. Moving forward, we derive insights from concept subspaces in terms of localized input (concept) maps, show how to quantify concept relevances and lastly, evaluate similarities and transferability between concepts. We empirically demonstrate the soundness of the proposed Sparse Subspace Clustering for Concept Discovery (SSCCD) method for a variety of different image classification tasks. This approach allows for deeper insights into the actual model behavior that would remain hidden from conventional input-level heatmaps.
Kernel machines have sustained continuous progress in the field of quantum chemistry. In particular, they have proven to be successful in the low-data regime of force field reconstruction. This is because many equivariances and invariances due to physical symmetries can be incorporated into the kernel function to compensate for much larger data sets. So far, the scalability of kernel machines has however been hindered by its quadratic memory and cubical runtime complexity in the number of training points. While it is known that iterative Krylov subspace solvers can overcome these burdens, their convergence crucially relies on effective preconditioners, which are elusive in practice. Effective preconditioners need to partially presolve the learning problem in a computationally cheap and numerically robust manner. Here, we consider the broad class of Nystrom-type methods to construct preconditioners based on successively more sophisticated low-rank approximations of the original kernel matrix, each of which provides a different set of computational trade-offs. All considered methods aim to identify a representative subset of inducing (kernel) columns to approximate the dominant kernel spectrum.
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