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An approximate projection onto the tangent cone to the variety of third-ordertensors of bounded tensor-train rank is proposed and proven to satisfy a betterangle condition than the one proposed by Kutschan (2019). Such an approximateprojection enables, e.g., to compute gradient-related directions in the tangentcone, as required by algorithms aiming at minimizing a continuouslydifferentiable function on the variety, a problem appearing notably in tensorcompletion. A numerical experiment is presented which indicates that, inpractice, the angle condition satisfied by the proposed approximate projectionis better than both the one satisfied by the approximate projection introducedby Kutschan and the proven theoretical bound.

An Approximate Projection onto the Tangent Cone to the Variety of Third-Order Tensors of Bounded Tensor-Train Rank