# Linear embeddings of low-dimensional subsets of a Hilbert space to $\mathbb{R}^m$

1 PANAMA - Parcimonie et Nouveaux Algorithmes pour le Signal et la Modélisation Audio
IRISA-D5 - SIGNAUX ET IMAGES NUMÉRIQUES, ROBOTIQUE, Inria Rennes – Bretagne Atlantique
Abstract : We consider the problem of embedding a low-dimensional set, M, from an infinite-dimensional Hilbert space, H, to a finite-dimensional space. Defining appropriate random linear projections, we propose two constructions of linear maps that have the restricted isometry property (RIP) on the secant set of M with high probability. The first one is optimal in the sense that it only needs a number of projections essentially proportional to the intrinsic dimension of M to satisfy the RIP. The second one, which is based on a variable density sampling technique, is computationally more efficient, while potentially requiring more measurements.
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Cited literature [10 references]

https://hal.inria.fr/hal-01116153
Contributor : Gilles Puy <>
Submitted on : Monday, June 8, 2015 - 2:17:23 PM
Last modification on : Thursday, January 7, 2021 - 4:30:08 PM

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• HAL Id : hal-01116153, version 2

### Citation

Gilles Puy, Mike E. Davies, Rémi Gribonval. Linear embeddings of low-dimensional subsets of a Hilbert space to $\mathbb{R}^m$. EUSIPCO - 23rd European Signal Processing Conference, Aug 2015, Nice, France. ⟨hal-01116153v2⟩

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