We consider a system of m linear equations in n variables Ax=b where A is a given m x n matrix and b is a given m-vector known to be equal to Ax' for some unknown solution x' that is integer and k-sparse: x' in {0,1}^n and exactly k entries of x' are 1. We give necessary and sufficient conditions for recovering the solution x exactly using an LP relaxation that minimizes l1 norm of x. When A is drawn from a distribution that has exchangeable columns, we show an interesting connection between the recovery probability and a well known problem in geometry, namely the k-set problem. To the best of our knowledge, this connection appears to be new in the compressive sensing literature. We empirically show that for large n if the elements of A are drawn i.i.d. from the normal distribution then the performance of the recovery LP exhibits a phase transition, i.e., for each k there exists a value m' of m such that the recovery always succeeds if m > m' and always fails if m < m'. Using the empirical data we conjecture that m' = nH(k/n)/2 where H(x) = -(x)log_2(x) - (1-x)log_2(1-x) is the binary entropy function.
We consider the system of linear equations in the real vector variable x:
(1) where A is a given real m × n matrix, b is a given vector in R m and x ∈ R n . Suppose it is known that the system has an underlying solution that is binary, i.e., ∃ x ∈ {0, 1} n such that b = Ax. We are interested in the conditions under which x can be recovered exactly and efficiently.
The above problem occurs in smart grids where we wish to retrieve the underlying phase connectivity from a time series of meter measurements [1]. Each customer household is connected to one of the three phases of a low-voltage transformer that distributes power to households. Both households and transformers have smart meters. Therefore for a series of time intervals, it is known as to how many watt-hours of power are sent out on each phase and how many are consumed by each customer. However, to which phase a customer is connected is not known. System (1) is the problem formulation for a single phase based on the principles of conservation of power. The columns of A hold the time series of meter measurements from customers, b holds the corresponding time series for a phase, and x determines if a customer is connected to that phase. Empirical data collected from real smart meters shows that measurements have sufficient variability over time and customers implying that A has full rank.
If m = n, the unique underlying binary solution to (1) is recovered as x = A -1 b. If m < n, system (1) has infinite real solutions and may have multiple binary solutions. When m = 1, the problem reduces to Subset-Sum problem, which is NP-hard. For m < n, even if a binary solution to (1) is given, checking if it is a unique solution is also NP-hard [2].
To circumvent these difficulties, Mangasarian et al. [3] first transform (1) to its equivalent Ay = d using y = e -2x where d = Ae -2b and e is a column vector of all ones. Then they give necessary and sufficient conditions for the uniqueness of an integer solution y ∈ {-1, 1} n to the following LP relaxation that minimizes the ∞ norm of y: min δ s.t. Ay = d, -δe ≤ y ≤ δe.
(2) A solution ȳ that is unique to LP (2) and is integer guarantees that (eȳ)/2 exactly recovers x.
The paper [3] also computes the probability that a randomly generated problem instance of LP (2) satisfies the uniqueness conditions. This gives a lower bound on the probability that (1) has a unique binary solution. For large n, a transition behavior is observed: the probability of uniqueness is almost 0 for m/n < 1/2 and almost 1 for m/n > 1/2.
In this work, we follow the approach of [3] and study the conditions under which a binary and ksparse x with exactly k non-zero entries is a unique solution to the following alternate LP relaxation that minimizes the 1 norm of x:
) so that (3) exactly recovers the solution x of (1). As in [3], we wish to find necessary and sufficient conditions for the uniqueness of a binary k-sparse solution and to compute the probability of the conditions getting satisfied on a random instance as a function of n, m, k. Donoho et al. [4] also consider recovery of sparse solutions to (1), albeit with a different notion of sparsity. They consider three LP relaxations of which two are relevant to our work:
min e T x s.t. Ax = b, x ≥ 0.
(5) Donoho et al.’s definition of sparsity is closely tied to the polytope defined by the constraints of the LP relaxation. A signal is considered k-sparse if it lies on a k-face of the constraint polytope. However in LP (3) a k-sparse binary signal does not lie on the k-face of its associated polytope. Therefore their techniques of counting faces of polytopes to compute uniqueness probabilities give us partial results, however they are not tight.
For example, in case of LP (4), x is considered k-sparse if it has nk entries either 0 or 1 and k entries in (0, 1). Any binary signal lies on a vertex of LP (4)’s constraints polytope 0 ≤ x ≤ 1 and has zero sparsity. Donoho et al. show that LP (4) requires m/n = 1/2 to recover a signal of zero sparsity (see figure 3 of [4], Q = I). This is not the strongest result possible because LP (3) can recover certain binary signals with m/n strictly less than 1/2. In case of LP (5), x is considered k-sparse if nk entries are 0 and k entries > 0. In this case although their definition of sparsity coincides with ours, the results for recovery are not tight. For m/n = 1/2, LP (5) recovers a binary signal of sparsity at most k = n/4 (see figure 3 of [4], Q = T ) while LP (3) can even recover a binary signal of sparsity k = n/2.
The rest of the paper is organized as follows. Section 2 gives necessary and sufficient conditions for a binary and k-sparse signal to be the unique solution of LP (3). Section 3 links the probability of uniqueness with the expected number of k-sets in a random set of points from R m . The latter problem is still open. Section 4 presents experimental results and compares the performance of LPs.
2 Conditions for a k-sparse x ∈ {0,
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