Remi Gribonval just reminded about this important paper that I somehow missed: Dictionary Identification - Sparse Matrix-Factorisation via L1-Minimisation by Remi Gribonval and Karin Schnass. The abstract reads:
This article treats the problem of learning a dictionary providing sparse representations for a given signal class, via $\ell_1$-minimisation. The problem can also be seen as factorising a $\ddim \times \nsig$ matrix $Y=(y_1 \gt ... y_\nsig), y_n\in \R^\ddim$ of training signals into a $\ddim \times \natoms$ dictionary matrix $\dico$ and a $\natoms \times \nsig$ coefficient matrix $\X=(x_1... x_\nsig), x_n \in \R^\natoms$, which is sparse. The exact question studied here is when a dictionary coefficient pair $(\dico,\X)$ can be recovered as local minimum of a (nonconvex) $\ell_1$-criterion with input $Y=\dico \X$. First, for general dictionaries and coefficient matrices, algebraic conditions ensuring local identifiability are derived, which are then specialised to the case when the dictionary is a basis. Finally, assuming a random Bernoulli-Gaussian sparse model on the coefficient matrix, it is shown that sufficiently incoherent bases are locally identifiable with high probability. The perhaps surprising result is that the typically sufficient number of training samples $\nsig$ grows up to a logarithmic factor only linearly with the signal dimension, i.e. $\nsig \approx C \natoms \log \natoms$, in contrast to previous approaches requiring combinatorially many samples.
I'll add this paper shortly to the big picture in compressive sensing page. Also found on arxiv:
Compressive Sensing Based Opportunistic Protocol for Throughput Improvement in Wireless Networks by Syed T. Qaseem, and Tareq Y. Al-Naffouri. The abstract reads:
A key feature in the design of any MAC protocol is the throughput it can provide. In wireless networks, the channel of a user is not fixed but varies randomly. Thus, in order to maximize the throughput of the MAC protocol at any given time, only users with large channel gains should be allowed to transmit. In this paper, compressive sensing based opportunistic protocol for throughput improvement in wireless networks is proposed. The protocol is based on the traditional protocol of R-ALOHA which allows users to compete for channel access before reserving the channel to the best user. We use compressive sensing to find the best user, and show that the proposed protocol requires less time for reservation and so it outperforms other schemes proposed in literature. This makes the protocol particularly suitable for enhancing R-ALOHA in fast fading environments. We consider both analog and digital versions of the protocol where the channel gains sent by the user are analog and digital, respectively.
Credit: Tamas Ladanyi, Pee Over Hungary, By the Ruins of Essegvar.
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