A variant of the Johnson–Lindenstrauss lemma for circulant matrices

Abstract

We continue our study of the Johnson-Lindenstrauss lemma and its connection to circulant matrices started in Hinrichs and Vybíral (in press) [7]. We reduce the bound on k from k = Ω(ε −2 log 3 n) proven there to k = Ω(ε −2 log 2 n). Our technique differs essentially from the one used in Hinrichs and Vybíral (in press) [7]. We employ the discrete Fourier transform and singular value decomposition to deal with the dependency caused by the circulant structure.

Compressed Fingerprint Matching and Camera Identification via Random Projections

“…However, the latter result is not sharp and recent works [15], [28] focused on improving the bound, closing some of the gap with respect to the result on Gaussian matrices. This highlights the slight reduction on performance due to the circulant structure with respect to a fully random matrix.…”

“…Then, the compressive matching system with real-valued measurements applies (14) to compute the random projections y = k of the test fingerprint and compares them with each column of the compressed dictionary A, namely a j = d j . On the other hand, in case of binary measurements the system applies (15) to compute the binary random projections y = sign( k ) of the test fingerprint and compares them with each column of the binarized compressed dictionary A, namely a j = sign( d j ). We now formally define the probabilities of the events introduced in the previous section.…”

“…Several results, both theoretical and experimental [15], [33], [34], suggest that partial circulant matrices are almost as effective as fully random Gaussian matrices, despite their structure and limited randomness. We compare the ROC obtained on the PoliTO database for Gaussian and circulant matrices, having the first row drawn as Gaussian i.i.d..…”

“…As to practical issues, the complexity of randomly projecting a large fingerprint is greatly reduced by employing partial circulant matrices [15], which are known to be almost as good as fully random matrices. Moreover, inspired both by the work of [12] and by recent results in compressed sensing literature [16], we propose a binary version of the compressed fingerprint that further reduces storage and computational requirements.…”

Suprema of Chaos Processes and the Restricted Isometry Property

“…Combining our result with the work [26] on the relation between the restricted isometry property and the Johnson-Lindenstrauss lemma, we also obtain an improved estimate for Johnson-Lindenstrauss embeddings arising from partial random circulant matrices; see also [24,47] for earlier work in this direction. THEOREM 1.2.…”