COMMUNITY DETECTION IN STOCHASTIC BLOCK MODELS VIA SPECTRAL METHODS Laurent Massoulié (MSR-Inria...
-
Upload
gervais-russell -
Category
Documents
-
view
222 -
download
1
Transcript of COMMUNITY DETECTION IN STOCHASTIC BLOCK MODELS VIA SPECTRAL METHODS Laurent Massoulié (MSR-Inria...
COMMUNITY DETECTION IN STOCHASTIC BLOCK MODELS VIA SPECTRAL METHODS
Laurent Massoulié (MSR-Inria Joint Centre, Inria)
Outline – remainder of the course
Control of eigen-elements’ perturbation Courant-Fisher min-max theorem Weyl’s inequalities
Bounding spectral norm of random noise matrices Trace method Matrix Bernstein inequalities Alon-Boppana theorem re. Ramanujan property
The tree reconstruction problem Branching number of a tree & Threshold for reconstruction From tree reconstruction to SBM reconstruction
Proof elements for modified spectral methods Matrix expansion formula « Local analysis »: quasi-deterministic growth
Outline – remainder of the course
Control of eigen-elements’ perturbation Courant-Fisher min-max theorem Weyl’s inequalities
Bounding spectral norm of random noise matrices Trace method Matrix Bernstein inequalities Alon-Boppana theorem re. Ramanujan property
The tree reconstruction problem Branching number of a tree & Threshold for reconstruction From tree reconstruction to SBM reconstruction
Proof elements for modified spectral methods Matrix expansion formula « Local analysis »: quasi-deterministic growth
Controlling perturbation of eigen-elements of Hermitian matrices
Where : observed; : signal; : perturbation
Perturbation Lemmaorder ,
1) (Weyl)
2) Let . IfThen for all i and for any normed eigenvector , there exists s.t.
Application to SBM
Spectrum of : spectrum of scaled by , hence
If there exist block-specific vectors such that for all
Clustering of spectral representatives follows ( eg take )
Another application of Courant-Fisher minimax theorem
Cauchy interlacing theorem:For Hermitian matrix , where : orthogonal projection on -dimensional space, then for all
Where : eigenvalues of viewed as operator on -dimensional space
Outline – remainder of the course
Control of eigen-elements’ perturbation Courant-Fisher min-max theorem Weyl’s inequalities
Bounding spectral norm of random noise matrices Trace method Matrix Bernstein inequalities Alon-Boppana theorem re. Ramanujan property
The tree reconstruction problem Branching number of a tree & Threshold for reconstruction From tree reconstruction to SBM reconstruction
Proof elements for modified spectral methods Matrix expansion formula « Local analysis »: quasi-deterministic growth
Outline – remainder of the course
Control of eigen-elements’ perturbation Courant-Fisher min-max theorem Weyl’s inequalities
Bounding spectral norm of random noise matrices Trace method Matrix Bernstein inequalities Alon-Boppana theorem re. Ramanujan property
The tree reconstruction problem Branching number of a tree & Threshold for reconstruction From tree reconstruction to SBM reconstruction
Proof elements for modified spectral methods Matrix expansion formula « Local analysis »: quasi-deterministic growth
Result à la Furedi-Komlos
centered independent with Then whp in probability as
e.g. in Gaussian case , :
Supremum of over unit sphere = typical value
A toy version illustrating the « trace method »
For centered independent with
For all , whp
Implies if for some fixed , A first sufficient condition for consistency of basic method
Another tool: Matrix Bernstein inequality [Tropp’10&’14]
For dimensional independent Hermitian matrices such that:, almost surely,Note and Then for all :
Hence:
Application
Show that in SBM, with high probability,
Deduce consistency of spectral method in SBM for signal strength
Key lemmas
Lemma 1 (« Master inequality »)
Lemma 2 (consequence of Lieb’s theorem):For independent Hermitian matrices , Then
Lemma 3: For hermitian such that and a.s.,
Result follows by proper choice of …
spectral separation properties “à la Ramanujan”
s-regular graph Ramanujan if
[Lubotzky-Phillips-Sarnak’88]
[Friedman’08]: random s-regular graph verifies whp
[Feige-Ofek’05]: for Erdős-Rényi graph and , then whp Also: . Result carries over to SBM