Mixed Hölder matrix discovery via wavelet shrinkage and Calderón–Zygmund decompositions

This paper concerns two related problems in the analysis of data matrices whose rows and columns are equipped with tree metrics. First is the problem of recovering a matrix that has been corrupted by additive noise. Under the assumption that the clean matrix exhibits a specific regularity condition, known as the mixed Hölder condition, we adapt the well-known Donoho–Johnstone wavelet shrinkage methods from classical nonparametric statistics to obtain estimators that are within a logarithmic factor of the minimax error rate with respect to mean squared error loss. The second part of this paper develops a theory of Besov spaces on products of tree geometries. We show that matrices with small Besov norm can be written as a sum of a mixed Hölder matrix and a matrix with small support. Such decompositions are known as Calderón–Zygmund decompositions and are of general interest in harmonic analysis. The decompositions we establish impose fewer conditions on the function with small support than previous decompositions of this type while maintaining the same guarantees on the mixed Hölder matrix. As such, they are applicable to a greater variety of matrices and should find use in many data organization problems. As part of our analysis, we provide characterizations of the underlying Besov spaces using wavelets and other multiscale difference operators that are analogous to those from the classical Euclidean theory.