Super-symmetric tensors - a higher-order extension of scatter matrices - are\nbecoming increasingly popular in machine learning and computer vision for\nmodelling data statistics, co-occurrences, or even as visual descriptors.\nHowever, the size of these tensors are exponential in the data dimensionality,\nwhich is a significant concern. In this paper, we study third-order\nsuper-symmetric tensor descriptors in the context of dictionary learning and\nsparse coding. Our goal is to approximate these tensors as sparse conic\ncombinations of atoms from a learned dictionary, where each atom is a symmetric\npositive semi-definite matrix. Apart from the significant benefits to tensor\ncompression that this framework provides, our experiments demonstrate that the\nsparse coefficients produced by the scheme lead to better aggregation of\nhigh-dimensional data, and showcases superior performance on two common\ncomputer vision tasks compared to the state-of-the-art.\n