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Consider finite-dimensional settings. Usual spectral theory decomposes a self-adjoint operators $A$ as a linear combination of orthogonal projections $\{P_i\}$ onto linear subspaces, e.g. $A = \sum_{i=1}^r \lambda_i P_i$.

Has there been any attempt to relax the linearity of the subspaces. For example, are there results where one tries to represent $A$ as the sum of projections onto convex sets (say convex cones)?

Here is a vaguely related (open-ended) problem: a version of nonnegative matrix factorization of $A$ attempts to write $A = \sum_{i=1}^r \lambda_i u_i u_i^T$ where $u_i \in \mathbb{R}_+$ are vectors with nonnegative coordinates (maybe with some additional constraints). Can this be related to the above problem in any way?

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