A Computational Theory of Robust Localization Verifiability in the Presence of Pure Outlier Measurements

We assume that the probabilities $p_{E}=\{p_{ij}\}_{(i,j)\in E}$ are known; as shown below in Theorem III.3, our results are valid independently of the support for $\mathcal{U}^{\pm}$ (as long as it is finite).

From this point on, subscripts with $V$ or $E$ refer to the vector obtained by stacking the specified quantity considered for all nodes or edges (e.g., $p_{E}=\operatorname{stack}(\{p_{ij}\}_{(i,j)\in E})$).

Given the relative pairwise measurements $t_{E}$ in the graph $G$, we aim to find and characterize all the embeddings that minimize the sum of all absolute residuals, i.e.,

	$\displaystyle\min_{\mathbf{X}_{V},\mathbf{x}_{1}=\mathbf{0}}\sum_{(i,j)\in E}\lVert\mathbf{x}_{j}-\mathbf{x}_{i}-t_{ij}\rVert_{1}$		(3)
	.

If we translate all the points in the embedding by a common translation, the cost (3) does not change, since the relative measurements also remain constant. Without loss of generality, we fix this translation ambiguity by choosing a global reference frame such that $\mathbf{x}_{1}^{\ast}=\mathbf{x}_{1}=\mathbf{0}_{d}$. Since we assumed that the graph is connected (Definition 1), fixing $\mathbf{x}_{1}$ alone is sufficient to fix the global translation. For simplicity’s sake, we keep $\mathbf{x}_{1}$ as a variable in the optimization problem (3) even though it is used to fix the global translational ambiguity.

We define as $\mathcal{X}^{\textrm{opt}}$ the set of local minimizers of (3). Since the objective function is convex (being the sum of convex functions), we have that $\mathcal{X}^{\textrm{opt}}$ is convex, and is exactly given by the set of global minimizers (see [4, Theorems 8.1, 8.3]). Moreover, using the fact that the value of $\mathbf{x}_{1}$ is fixed and that the graph is connected, it is possible to show that the objective function in (3) is radially unbounded, and therefore the set $\mathcal{X}^{\textrm{opt}}$ is compact. In fact, since (3) can be rewritten as a Linear Program (LP, see below), $\mathcal{X}^{\textrm{opt}}$ either reduces to a single point, or is a polyhedron with a finite number of corners (we use this term instead of vertex as a distinction from the individual elements of $V$).

If $E_{\epsilon}=\emptyset$, then $t_{E}$ is identical to the true measurements, and the solution of (3) would be equal to the ground truth embedding $\mathbf{X}^{\ast}_{V}$. However, since (3) is a robust optimization problem, the optimum value could still correspond to $\mathbf{X}^{\ast}_{V}$ even in the presence of outliers ($E_{\epsilon}\neq\emptyset$). In the latter case, however, there could be multiple minimizers all giving the same value of the $\ell_{1}$ objective. We start formalizing the situation with the following.

Note that, according to the definitions, uniquely verifiable problems are also verifiable.

In [31, Theorem 2], the authors also introduce the concept of verifiable edge and verifiable graph; however, that work considers only the case of a single outlier ($\lvert E_{\epsilon}^{\pm}\rvert=1$). In this work, we generalize the same notion to more general cases.

We first perform a change of variable so that the true embedding corresponds to the point at the origin. More in detail, we define a set of new variables $\mathbf{X}^{\prime}_{V}$ such that

$$\mathbf{X}_{V}^{\prime}=X_{V}-\mathbf{X}^{\ast}_{V},$$

(4)

i.e., for each ${i\in V}$ we replace $\mathbf{x}_{i}$ by $\mathbf{x}^{\prime}_{i}+\mathbf{x}_{i}^{\ast}$. If $\mathbf{X}^{\ast}$ is an optimal point for (3), then $\mathbf{X}^{\prime}=\mathbf{0}_{\lvert V\rvert}$ is a minimizer for the following transformed problem:

$\displaystyle\min_{\mathbf{x}^{\prime}_{V},\mathbf{x}^{\prime}_{1}=0}\sum_{(i,j)\in E}\lVert(\mathbf{x}^{\prime}_{j}+\mathbf{x}_{j}^{\ast})-(\mathbf{x}^{\prime}_{i}+\mathbf{x}_{i}^{\ast})-(\mathbf{x}^{*}_{j}-\mathbf{x}^{*}_{i}+\epsilon_{ij})\rVert_{1},$

(5)

which reduces to

$\displaystyle\min_{\mathbf{x}^{\prime}_{V},\mathbf{x}^{\prime}_{1}=0}\sum_{(i,j)\in E}\lVert\mathbf{x}^{\prime}_{j}-\mathbf{x}^{\prime}_{i}-\epsilon_{ij}\rVert_{1}.$

(6)

By inspecting (6), we can deduce the following:

The practical implication of Lemma III.1 is that we can reason about the verifiability of a problem independently from the specific true positions of nodes. To simplify our discussion, for the remainder of the paper and without loss of generality we use $\mathbf{x}$ instead of $\mathbf{x}^{\prime}$.

The $\ell_{1}$-norm $\lVert\cdot\rVert_{1}\colon d\rightarrow$ in the optimization objective can be decomposed into sums of absolute values across dimensions, i.e., (6) becomes

$\displaystyle\min_{\mathbf{X}_{V},[\mathbf{x}_{1}]_{k}=0}$

$\displaystyle\sum_{k=1}^{d}\sum_{(i.j)\in E}\bigl{\lvert}[\mathbf{x}_{j}]_{k}-[\mathbf{x}_{i}]_{k}-[\epsilon_{ij}]_{k}\bigr{\rvert},$

(7)

where $[v]_{k}$ denotes the $k$-th element of a vector $v\in d$. The minimization problem (7) can then be decomposed into $d$ separate optimization problems, each one with a solution set $[\mathcal{X}^{\textrm{opt}}]_{k}$, $k\in\{1,\ldots,d\}$, and each one corresponding to a 1-D localization problem of the form

$\displaystyle\min_{x_{V},x_{1}=0}\sum_{(i.j)\in E}\mid x_{j}-x_{i}-\epsilon_{ij}\mid.$

(8)

We postpone to Section IV-D the discussion of how to combine the results of our analysis from the different dimensions; until that section, we exclusively focus on the 1-D version of the problem.

In this section, we transform (8) into the equivalent standard Linear Program (LP) form, with a linear cost function subject to linear inequality constraints, and compute its dual. This will allow us to arrive to the conclusion that the exact magnitude of the outliers is not important in terms of verifiability, and only the signed outlier support matter.

We first introduce variables

$$Z_{ij}=\lvert x_{j}-x_{i}-\epsilon_{ij}\rvert,\;\;\forall(i,j)\in E,$$

(9)

to push the cost function into the constraints.

	$\displaystyle\min_{Z_{E},x_{V},x_{1}=0}$	$\displaystyle\sum_{(i,j)\in E}Z_{ij}$		(10a)
	subject to	$\displaystyle x_{j}-x_{i}-\epsilon_{{ij}}\leq Z_{{ij}},$		(10b)
		$\displaystyle-(x_{j}-x_{i}-\epsilon_{{ij}})\leq Z_{{ij}},$		(10c)
		$\displaystyle Z_{{ij}}\geq 0,$		(10d)
		$\displaystyle\forall{i\in V},(i,j)\in E.$

Next, in order to obtain a standard LP form, all variables must be non-negative. We therefore split each variable $x_{i}$ into the summation of two non-negative variables,

$$x_{i}=x^{+}_{i}-x^{-}_{i},\;x^{+}_{i},x^{-}_{i}\geq 0.$$

(11)

Finally, we change the inequality constraints into equality constraints by introducing the slack variables $S^{+}_{E},S^{-}_{E}$:

	$\displaystyle\min_{Z,x,x_{1}=0}$	$\displaystyle\sum_{(i.j)\in E}Z_{ij},$		(12a)
	subject to	$\displaystyle x^{+}_{j}-x^{-}_{j}-(x^{+}_{i}-x^{-}_{i})-\epsilon_{{ij}}+S^{+}_{{ij}}=Z_{{ij}},$		(12b)
		$\displaystyle-(x^{+}_{j}-x^{-}_{j}-(x^{+}_{i}-x^{-}_{i})-\epsilon_{{ij}})+S^{-}_{{ij}}=Z_{{ij}},$		(12c)
		$\displaystyle x^{+}_{i},x^{-}_{i},S^{+}_{ij},S^{-}_{ij},Z_{ij}\geq 0,$		(12d)
		$\displaystyle\forall{i\in V},\;(i,j)\in E.$

We can also form the dual optimization problem of (12),

	$\displaystyle\max_{P_{ij}^{+},P_{ij}^{-}}$	$\displaystyle\sum_{(i,j)\in E}{\epsilon_{ij}(P_{ij}^{+}-P_{ij}^{-})},$		(15a)
	subject to	$\displaystyle\sum_{j,(j,i)\in E}(P_{ji}^{+}-P_{ji}^{-})-\sum_{j,(i,j)\in E}(P_{ij}^{+}-P_{ij}^{-})=0,\;$		(15b)
		$\displaystyle-P_{ij}^{+}-P_{ij}^{-}\leq 1,$		(15c)
		$\displaystyle P_{ij}^{+},P_{ij}^{-}\leq 0,$		(15d)
		$\displaystyle\forall{i\in V},(i,j)\in E,$

where $P^{+}_{ij}$ is the dual variable associated to constraint (12b), and $P^{-}_{ij}$ is the dual variable associated to constraint (12c).

These remarks allow us to prove the following.

Combining lemmata III.1 and III.2 we have the following:

Technically speaking, the proof above does not cover the case of unique verifiability, in the sense that the they do not exclude the case where a verifiable problem might become uniquely verifiable after rescaling (or viceversa). We are investigating this issue in our current work.

In this section, we discuss how the dual simplex algorithm can be used to compute the verifiability of a given problem. As a result of the previous section, for our analysis, the values of $\epsilon_{E}$ can be choosen randomly, as long as they have the correct edge support $E_{\varepsilon}^{\pm}$. We start by rewriting the LP (12) in matrix form:

		$\displaystyle\min_{q}$		$\displaystyle c^{\mathrm{T}}q$		(20)
		subject to		$\displaystyle Aq=b$
		$\displaystyle q\geq 0.$

The vector $c=\operatorname{stack}(\mathbf{0}_{2\lvert V\rvert},\mathbf{1}_{\lvert E\rvert},\mathbf{0}_{2\lvert E\rvert})$ contains the set coefficients in the cost function, while $A\in\{0,1,-1\}^{2\lvert E\rvert\times(2\lvert V\rvert+3\lvert E\rvert)}$, and $b=\left[\begin{smallmatrix}1\\ -1\end{smallmatrix}\right]\otimes\epsilon_{E}$ defines the constraints (where $\otimes$ denotes the Kronecker’s product). Finally, the vector $q=\operatorname{stack}(x_{V}^{+},x_{V}^{-},Z_{E},S_{E}^{+},S_{E}^{-})\in\mathbb{R}^{2\lvert V\rvert+3\lvert E\rvert}$ contains the decision variables.

Given the standard form of the optimization problem (20), we can use the dual simplex algorithm [5] to find all the corners of the set of minimizers $\mathcal{X}^{\textrm{opt}}$. The algorithm and its application to our problem are summarized next.

The dual simplex method is based on the following concepts:

1. 1.

   Basic variables (BVs): a subset of variables ($q_{B}$), that, together with the constraints, defines the current candidate solution in the algorithm. Non-basic variables (NBV) are always zero.

2. 2.

   Simplex tableau: a $(2\lvert E\rvert+1)\times(2|V|+3|E|-1)$ array where

   • •

     The zeroth column represents the value of the set of basic variables ($q_{B}$). It is initialized with the vector $b$.

   • •

     The zeroth row contains the reduced costs, which are defined as the penalty cost for introducing one unit of the variable $q_{i}$ to the cost. These are initialized with the vector $c$.

   • •

     Columns one to $2(|V|-1)+3|E|$ are each one associated with one variable, where we excluded the columns corresponding to $x^{+}_{1},x^{-}_{1}$, since $x_{1}$ is fixed in the optimization. These columns are initialized with the matrix $A$.

For our initial estimated solution, we set all variables to zero except the slack variables; as a result, our initial BVs correspond to the set of slack variables, while the rest are NBVs. See Fig. 1 for an illustration of the initial tableau.

A typical iteration starts with some basic variables containing negative elements, and all reduced costs non-negative. For instance, in Fig. 1, the initial BVs are selected to be slack variables where $S^{+}_{ij}=\epsilon_{ij}$ and $S^{-}_{ij}=-\epsilon_{ij}$, hence, there are some negative initial BVs, while all reduced costs are non-negative (as all elements of vector $c$ are non-negative). These two properties are always maintained by the algorithm from one iteration to the next.

The iterations of the algorithm then follow these steps:

1. 1.

   Check for termination due to optimality: Examine the elements of zeroth column (which constitutes the basic set). If all of them are non-negative, we have an optimal basic solution and the algorithm terminates.

2. 2.

   Choose pivot row: Find some $\nu$ such that $[q_{B}]_{\nu}<0$.

3. 3.

   Check for termination due to unbounded solution: Considering the $\nu$-th row of the tableau, with elements $r_{1},\ldots,r_{2(\lvert V\rvert-1)+3\lvert E\rvert}$, if all the elements of the row are non-negative, the optimal dual cost is $+\infty$ and algorithm terminates. Since the set of minimizers $\mathcal{X}^{\textrm{opt}}$ in our problem is bounded (see Section II-D), this condition is never encountered in our application.

4. 4.

   Choose pivot column: For each $i$ such that $r_{i}<0$, compute the ratio $\bar{c_{i}}/|r_{i}|$ where $\bar{c_{i}}$ is the reduced cost of variable $q_{i}$ and let $j$ be the index of a column that correspond to the smallest ratio.

5. 5.

   Pivoting: Remove the variable $[q_{B}]_{\nu}$ from the basis, and have variable $q_{j}$ take its place. Add to each row of the tableau a multiple of the $\nu$-th row (pivot row) so that $r_{j}$ (the pivot element) becomes $1$ and all other entries of the pivot column become $0$. As a result, the total cost is reduced by the reduced cost $\bar{c}_{j}$.

6. 6.

   Repeat the algorithm from step 2 until all elements of $q_{B}$ are non-negative or the algorithm otherwise terminates.

After solving the simplex tableau, we get the basic optimal solution, which contains non-negative elements, together with non-negative reduced costs. The solution of the dual simplex algorithm is an optimal solution for (20), and is a corner point of the feasible region (Theorem 2.3, [5]). If we have multiple optimal solutions (i.e., $\mathcal{X}^{\textrm{opt}}$ is not a singleton), there will be multiple other corners with the same cost.

Hence, it is of interest to computationally enumerate all the corners of $\mathcal{X}^{\textrm{opt}}$, as discussed next.

The LP problem 20 can have multiple optimal solutions only when two conditions are met [2]:

1. 1.

   There exists a non-basic variable with zero reduced cost. Pivoting this variable into the basis would not change the value for the cost function.

2. 2.

   There exists a degenerate basic solution, i.e. some basic variables are equal to zero.

If the two conditions above are met, the corners in $\mathcal{X}^{\textrm{opt}}$ can be enumerated using a depth first search [26]:

1. 1.

   Prepare a queue $Q$ of corners to visit, with the corresponding tableau, and initialize it with the current solution found by the dual simplex algorithm,

2. 2.

   For each corner in $Q$ and its associated tableau,

   1. (a)

      Choose $C_{col}$ as the set of columns associated to non-basic variables with zero reduced cost, for all $j\in C_{col}$,

      1. i.

         Choose $C_{row}$ as the set of elements of the $j$-th pivot column which are positive,

      2. ii.

         For $i\in C_{row}$, we perform the pivoting, so that the pivot element in $i$-th row and $j$-th column becomes $1$ and all other entries of the pivot column become $0$,

      3. iii.

         Add the current corner to the queue $Q$, if is not in it already,

3. 3.

   Go to step 2 until the queue $Q$ is empty.

There are three cases for the set of optimal solutions, $\mathcal{X}^{\textrm{opt}}$:

1. 1.

   Uniquely verifiable solution: Pivoting new variables to the basis does not result in new corner point; we therefore have a unique optimal solution $\mathcal{X}^{\textrm{opt}}=\{\mathbf{0}_{V}\}$, and from (4) we conclude that the resulting embedding is congruent to the ground truth.

2. 2.

   Verifiable (non-unique) solution: We have multiple optimal solutions, including the origin ($\mathbf{0}_{V}\in\mathcal{X}^{\textrm{opt}}$); hence, there are multiple optimal embeddings, with one of them being congruent to the ground truth.

3. 3.

   Non-verifiable: In this case, $\mathbf{0}_{V}\notin\mathcal{X}^{\textrm{opt}}$, and the ground truth embedding is not an optimal solution.

In Section III-B, we reduced one $d$-dimensional optimization problem of the form (6) to $d$ 1-D optimization problems of the form (8). Now, we need to combine the optimal solutions of all dimensions to characterize the $d$-dimensional optimal solution. Let $[\mathcal{X}^{\textrm{opt}}]_{k}$ represents the set of optimal solutions for the LP (10) of dimension $k$. The value of the cost function (10) is the same for all corner points in $[\mathcal{X}^{\textrm{opt}}]_{k}$. Due to this fact, we can pick a 1-D corner point from each set $[\mathcal{X}^{\textrm{opt}}]_{k}$, $k\in\{1,\ldots,d\}$, and combine them to build a $d$-dimensional corner point:

$$x^{opt}=\operatorname{stack}(X^{opt}_{1},\ldots,X^{opt}_{d}),\;X^{opt}_{k}\in[\mathcal{X}^{\textrm{opt}}]_{k}.$$

(21)

Let $\lvert[\mathcal{X}^{\textrm{opt}}]_{k}\rvert$ represents the cardinality of the set $[\mathcal{X}^{\textrm{opt}}]_{k}$; then, we have $N=\prod_{k=1}^{d}\bigl{\lvert}[\mathcal{X}^{\textrm{opt}}]_{k}\bigr{\rvert}$ $d$-dimensional corner points. To have a unique verifiable graph, we therefore need all the individual 1-D problems to be also unique verifiable, i.e. $\lvert[\mathcal{X}^{\textrm{opt}}]_{k}\rvert=1$ for all $k\in\{1,\ldots,d\}$.

If for all corners a subset of components $V^{\prime}$ in the solution are always zero (i.e., $[X^{opt}_{k}]_{V}^{\prime}=0$ for all $k$), then the position of those particular nodes, and all their relative positions, are congruent to the true embedding. As a consequence, also all their relative costs are the same. Hence, while the entire problem $G,E^{\pm}_{\epsilon}$ is not verifiable, the sub-problem $G^{\prime},E_{\epsilon}^{\prime\pm}$, where $G^{\prime}=(V^{\prime},E^{\prime})$, $E^{\prime}=\{(i,j)\in E:i,j\in V^{\prime}\}$ is verifiable. We call the maximal connected components of $G^{\prime}$ defined in this way the maximal verifiable components of $G$.