Tensor Representations for Action Recognition

Human actions in video sequences are characterized by the complex interplay between spatial features and their temporal dynamics. In this paper, we propose novel tensor representations for compactly capturing such higher-order relationships between visual features for the task of action recognition. We propose two tensor-based feature representations, viz. (i) <italic>sequence compatibility kernel</italic> (SCK) and (ii) <italic>dynamics compatibility kernel</italic> (DCK). SCK builds on the spatio-temporal correlations between features, whereas DCK explicitly models the action dynamics of a sequence. We also explore generalization of SCK, coined SCK<inline-formula><tex-math notation="LaTeX">$\;\oplus$</tex-math><alternatives><mml:math><mml:mrow><mml:mspace width="0.277778em"/><mml:mo>⊕</mml:mo></mml:mrow></mml:math><inline-graphic xlink:href="koniusz-ieq1-3107160.gif"/></alternatives></inline-formula>, that operates on subsequences to capture the local-global interplay of correlations, which can incorporate multi-modal inputs e.g., skeleton 3D body-joints and per-frame classifier scores obtained from deep learning models trained on videos. We introduce linearization of these kernels that lead to compact and fast descriptors. We provide experiments on (i) 3D skeleton action sequences, (ii) fine-grained video sequences, and (iii) standard non-fine-grained videos. As our final representations are tensors that capture higher-order relationships of features, they relate to co-occurrences for robust fine-grained recognition (Lin, 2017), (Koniusz, 2018). We use higher-order tensors and so-called Eigenvalue Power Normalization (EPN) which have been long speculated to perform spectral detection of higher-order occurrences (Koniusz, 2013), (Koniusz, 2017), thus detecting fine-grained relationships of features rather than merely count features in action sequences. We prove that a tensor of order <inline-formula><tex-math notation="LaTeX">$r$</tex-math><alternatives><mml:math><mml:mi>r</mml:mi></mml:math><inline-graphic xlink:href="koniusz-ieq2-3107160.gif"/></alternatives></inline-formula>, built from <inline-formula><tex-math notation="LaTeX">$Z_*$</tex-math><alternatives><mml:math><mml:msub><mml:mi>Z</mml:mi><mml:mo>*</mml:mo></mml:msub></mml:math><inline-graphic xlink:href="koniusz-ieq3-3107160.gif"/></alternatives></inline-formula> dimensional features, coupled with EPN indeed detects if at least one higher-order occurrence is ‘projected’ into one of its <inline-formula><tex-math notation="LaTeX">$\binom{Z_*}{r}$</tex-math><alternatives><mml:math><mml:mfenced separators="" open="(" close=")"><mml:mfrac linethickness="0pt"><mml:msub><mml:mi>Z</mml:mi><mml:mo>*</mml:mo></mml:msub><mml:mi>r</mml:mi></mml:mfrac></mml:mfenced></mml:math><inline-graphic xlink:href="koniusz-ieq4-3107160.gif"/></alternatives></inline-formula> subspaces of dim. <inline-formula><tex-math notation="LaTeX">$r$</tex-math><alternatives><mml:math><mml:mi>r</mml:mi></mml:math><inline-graphic xlink:href="koniusz-ieq5-3107160.gif"/></alternatives></inline-formula> represented by the tensor, thus forming a Tensor Power Normalization metric endowed with <inline-formula><tex-math notation="LaTeX">$\binom{Z_*}{r}$</tex-math><alternatives><mml:math><mml:mfenced separators="" open="(" close=")"><mml:mfrac linethickness="0pt"><mml:msub><mml:mi>Z</mml:mi><mml:mo>*</mml:mo></mml:msub><mml:mi>r</mml:mi></mml:mfrac></mml:mfenced></mml:math><inline-graphic xlink:href="koniusz-ieq6-3107160.gif"/></alternatives></inline-formula> such ‘detectors’.