Random Frame Decompositions from Weighted Residual Flows
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866909979839037440 |
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| author | Tian, James |
| author_facet | Tian, James |
| contents | We study the evolution of a positive operator under weighted residual maps determined by a finite family of orthogonal projections. Iterating these maps along the rooted tree of multi-indices produces a "weighted residual energy tree", together with natural path measures obtained by normalizing the dissipated energy or trace at each step. Under a quantitative coverage condition on the projections, we show that along almost every branch the residuals converge strongly to zero and the dissipated pieces admit a rank-one decomposition that reconstructs the initial operator. In the special case where the initial operator is the identity on a subspace, this yields almost surely a random Parseval frame generated intrinsically by the weighted residual dynamics. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_00349 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Random Frame Decompositions from Weighted Residual Flows Tian, James Functional Analysis Primary: 42C15, secondary: 47A63, 47B65 We study the evolution of a positive operator under weighted residual maps determined by a finite family of orthogonal projections. Iterating these maps along the rooted tree of multi-indices produces a "weighted residual energy tree", together with natural path measures obtained by normalizing the dissipated energy or trace at each step. Under a quantitative coverage condition on the projections, we show that along almost every branch the residuals converge strongly to zero and the dissipated pieces admit a rank-one decomposition that reconstructs the initial operator. In the special case where the initial operator is the identity on a subspace, this yields almost surely a random Parseval frame generated intrinsically by the weighted residual dynamics. |
| title | Random Frame Decompositions from Weighted Residual Flows |
| topic | Functional Analysis Primary: 42C15, secondary: 47A63, 47B65 |
| url | https://arxiv.org/abs/2601.00349 |