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We introduce probabilistic extensions of classical deterministic measures of algebraic complexity of a tensor, such as the rank and the border rank. We show that these probabilistic extensions satisfy various natural and algorithmically serendipitous properties, such as submultiplicativity under taking of Kronecker products. Furthermore, the probabilistic extensions enable improvements over their deterministic counterparts for specific tensors of interest, starting from the tensor 2, 2, 2 that represents 2 × 2 matrix multiplication. While it is well known that the (deterministic) tensor rank and border rank satisfy rk 2, 2, 2 = 7 and rk 2, 2, 2 = 7 * The research leading to these results has received funding from the European Research Council under the European Union's Seventh Framework Programme (FP/2007(FP/ -2013) / ERC Grant Agreement 338077 "Theory and Practice of Advanced Search and Enumeration". We gratefully acknowledge the use of computational resources provided by the Aalto Science-IT project at Aalto University.

We derandomize Valiant’s (J ACM 62, Article 13, 2015) subquadratic-time algorithm for finding outlier correlations in binary data. This demonstrates that it is possible to perform a deterministic subquadratic-time similarity join of high dimensionality. Our derandomized algorithm gives deterministic subquadratic scaling essentially for the same parameter range as Valiant’s randomized algorithm, but the precise constants we save over quadratic scaling are more modest. Our main technical tool for derandomization is an explicit family of correlation amplifiers built via a family of zigzag-product expanders by Reingold et al. (Ann Math 155(1):157–187, 2002). We say that a function $$f:\{-1,1\}^d\rightarrow \{-1,1\}^D$$ f : { - 1 , 1 } d → { - 1 , 1 } D is a correlation amplifier with threshold $$0\le \tau \le 1$$ 0 ≤ τ ≤ 1 , error $$\gamma \ge 1$$ γ ≥ 1 , and strength p an even positive integer if for all pairs of vectors $$x,y\in \{-1,1\}^d$$ x , y ∈ { - 1 , 1 } d it holds that (i) $$|\langle x,y\rangle |<\tau d$$ | ⟨ x , y ⟩ | < τ d implies $$|\langle f(x),f(y)\rangle |\le (\tau \gamma )^pD$$ | ⟨ f ( x ) , f ( y ) ⟩ | ≤ ( τ γ ) p D ; and (ii) $$|\langle x,y\rangle |\ge \tau d$$ | ⟨ x , y ⟩ | ≥ τ d implies $$\left (\frac{\langle x,y\rangle }{\gamma d}\right )^pD \le \langle f(x),f(y)\rangle \le \left (\frac{\gamma \langle x,y\rangle }{d}\right )^pD$$ ⟨ x , y ⟩ γ d p D ≤ ⟨ f ( x ) , f ( y ) ⟩ ≤ γ ⟨ x , y ⟩ d p D .

We address the problem of estimating three head pose angles in sign language video using the Pointing04 data set as training data. The proposed model employs facial landmark points and Support Vector Regression learned from the training set to identify yaw and pitch angles independently. A simple geometric approach is used for the roll angle. As a novel development, we propose to use the detected skin tone areas within the face bounding box as additional features for head pose estimation. The accuracy level of the estimators we obtain compares favorably with published results on the same data, but the smaller number of pose angles in our setup may explain some of the observed advantage.We evaluated the pose angle estimators also against ground truth values from motion capture recording of a sign language video. The correlations for the yaw and roll angles exceeded 0.9 whereas the pitch correlation was slightly worse. As a whole, the results are very promising both from the computer vision and linguistic points of view.