An alternative formulation using effective resistance

In this section we offer an alternative approach to obtain the inequality constraint (6.9) through electrical networks and the concept of effective resistance. Using this formulation, we then prove a lemma that shows that the inequality constraint enforces connected through a simple rounding of M .

We shortly introduce the concept of effective resistance in the electrical network interpretation of graphs. In contrast to s-t flow that interprets edge weights on a graph as flow capacities, electrical flow considers edge weights wij as the conductance

(inverse resistance) 1

rij between two nodes i, j.

Define the pseudoinverse of a Laplacian L+₌ Pn

i=2 1 λiviv > i where L = Pn i=2λiviv>i

is its eigendecomposition, and we note that the minimum eigenvalue λ1 corresponding to all 1 eigenvector is zero. Also note that LG[M ] denotes the Laplacian of the subgraph

with adjacency matrix A  M ( is the elementwise/Hadamard matrix product)2 _and let L+_{G[M ]} denote its pseudoinverse.

Defining a vector of directional current flows into/out of nodes with f (e.g. where positive elements are currents into the node and vice versa), the relationship between the vector of voltages v and f in an electrical circuit graph with resistances rij are

given by the relation v = L+_{f . This fact follows from Kirchoff’s current and voltage} laws (Vishnoi, 2012). The effective resistance Rab between any two nodes a, b is

then defined by the voltage difference va − vb = v>(ea− eb) when unit current is

flowing between two nodes such that f = ea − eb (or equivalently, inverse of the

current flow with unit voltage difference between a and b). We then have the identity Rab = (ea− eb)>v = (ea− eb)>L+(ea− eb).

2_{Corresponding to the original graph with edge weights scaled by M}

Given a root/anchor node r ∈ V that is assumed to be contained in the subgraph

S, consider the case where a current flow of diMii is present between r and i ∈ V on

the induced graph with Laplacian LG[M ]. Letting mi = diMii(er− ei), the voltage

vector vi _{corresponding to these inputs is given by v}i _{= L}+

G[M ]mi. We then obtain the

resistance between nodes r and i for input current diMii with the identity vri − vii =

(er− ei)L+_{G[M ]}mi = _d 1

iMiim

>

i L

+

G[M ]mi, which we intend to be finite for i ∈ S (and thus

Mii > 0) if GS is connected. For a conductance threshold τ , we would thus like to

impose the constraint

m>_i L+_{G[M ]}mi ≤

diMii

τ , (6.10)

for all i ∈ V , where the scaling with Mii serves to restrict the constraint to only the

nodes i ∈ S. We also remark that the constraint (6.10) is independent of uniform multiplicative scaling of M , thus constraining M to unit trace does not affect the choice of τ .

We next unify the n different resistance constraints to a single PSD constraint. Define the 2n × 2n matrix AM as

AM = LG[M ] LStar(r)_{[M ]} L_Star(r)_{[M ]} 1 τLStar(r)_{[M ]} ! ,

for which we define our connectivity constraint to be AM  0. For i 6= r consider the

submatrix Ai formed by the top-left n × n submatrix (i.e. LG[M ]) and the (n + i)-th

row and column, which results in

Ai =

LG[M ] m>i

mi diM_τii

!

.

Since AM  0 it implies that any such submatrix satisfies Ai  0 and through the

Schur complement condition that is equivalent to condition (6.10). We have thus shown that the condition AM  0 encapsulates the pairwise effective resistance constraints

where with currents mr ,Pj6=rmj applied to the nodes and a convex combination of

voltage differences, m>_rL+_{G[M ]}mr is being compared to

P

j6=rMjj

τ .

Finally, we can again utilize the Schur complement condition for positive semidefi- niteness and simplify the PSD condition AM  0 on the 2n × 2n matrix to a PSD

condition on an n × n matrix, which directly leads to inequality (6.9) when we replace τ with γ₂2.

Through this formulation of the connectivity constraint, we can state the following lemma which guarantees the connectivity of the rounded solutions of our optimization formulation.

Lemma 6.3.3. For any γ > 0 and M that is a feasible solution of (6.11), the subgraph

G_Sˆ for subset ˆS = {i ∈ V : Mii> 0} is connected.

Proof. Assume there exists i ∈ ˆS such that there exists no path between r and i, i.e.,

the subgraph G_Sˆ is disconnected. Since Qγ(M )  0, it follows that AM  0 and thus

Ai  0 for τ = γ

2

2 . Through the Schur complement lemma and (6.10) we have that v_ri − vi

i ≤ 1τ, i.e., that the voltage difference between nodes r and i is finite when a

current flow of diMii is applied that is non-zero (since Mii> 0). This voltage difference

is computed for the graph G with edge weights given by Mij. It is easy to see that if

this voltage difference is finite, the voltage difference in the original graph G is also finite since Mij are bounded. However a contradiction arises because there exists no

path between i and r in G and thus the effective resistance Rri = (er− ei)>L+(er− ei)

is infinite.

6.3.3 Primal and dual formulations

To conclude this section, we restate our relaxation and introduce some shorthand notation for its constraints. Let Qγ(M )  0 be our relaxed connectedness constraint,

i.e., Qγ(M ) = LG[M ]− γ

2

2 LStar(r)[M ]. Then our relaxation is given by

max

M ∈∆n,M ≥0

We now write the SDP dual of our relaxation. We consider a scalar α as the Lagrange multiplier corresponding to constraint I · M = 1, and matrices Y, Z ∈ Rn×n _corre-

sponding to Qγ(M )  0 and M ≥ 0 respectively. Let Pγ(Y ) =P(i,j)∈E(Lij · Y )eij −

γP

i∈V di(Lri· Y )eiibe the transpose of the constraint Qγ, i.e., Pγ(Y ) · M = Qγ(M ) · Y .

The dual can then be written as

min α s.t. C + Pγ(Y ) + Z  αI

α ≥ 0, Y  0, Z ≥ 0

At this point, we notice an important feature of our formulation. We have C ≥ 0, by definition for the elevated mean problem with nonnegative signal values and with high probability for correlation detection. Then the term C + Pγ(Y ) in the dual constraint

is the sum of a nonnegative matrix plus a diagonal matrix. By the Perron-Frobenius theorem, the top eigenvector has nonnegative components and we can assume w.l.o.g. that Z = 0. We will use the same reasoning in the next section to show that we do not need to explicitly enforce the n2 _{element-wise nonnegativity constraints corresponding} to M ≥ 0, as our dual formulation will automatically yield such solutions.