Considerable amounts of high-dimensional signals are continuously being acquired, whether audio, images, videos, or specialized signals for example in geophysics or medicine. Automatic classification and retrieval is strongly needed to analyze and access these massive data sets, but current algorithms often produce too many errors. For high-dimensional signals, supervised classification algorithms are typically applied to reduced ``feature vectors''. These feature representations are specialized for each signal modality, for example speech, music, images, videos or seismic signals. This proposal aims at unifying these approaches to improve classification performances, by developing a general mathematical and algorithmic framework to optimize representations for classification. Classification errors result from representations which are not sufficiently informative or which maintain too much variability. The central challenge is to understand how to construct stable, informative invariants, while facing progressively more complex sources of variability. The first task concentrates on invariants to the action of finite groups including translations, rotations and scalings, while preserving stability to deformations. The second task addresses unsupervised representation learning from training data. The third task explores stable representations of invariant geometric signal structures, which is an outstanding problem.These challenges involve building new mathematical tools in harmonic and wavelet analysis, geometry and statistics, in close interaction with numerical algorithms. Classification applications to audio, images, video signals or geophysical signals are expected to serve as a basis for groundbreaking technological advances.