Chiara Cammarota - King's College London Tensor PCA is a prototypical, particularly hard, high-dimensional estimation problem. In this talk I will discuss how it can be used to reveal, by means of statistical mechanics approaches, the details of the intimate connection between the geometry of the high dimensional cost landscape and the performances of gradient descent based algorithms. I will also show how the added knowledge about the cost landscape allows to devise both specific and potentially general strategies to substantially increase the performances of gradient descent based algorithms.