Robust Localization in Wireless Sensor Networks Via Low-Rank and Sparse Matrix Decomposition

Robust Localization in Wireless Sensor Networks via Low-rank and Sparse Matrix Decomposition

Beatrice Rossi, Marco Patanè, Pasqualina Andrea Fusiello Fragneto DPIA- University of Udine AST Lab - STMicroelectronics [email protected] [email protected] [email protected] [email protected]

ABSTRACT ad-hoc network, which can also be mobile, depending on the appli- This paper proposes a robust matrix completion method to solve cation. In this work we will consider both cases. the localization problem in wireless sensor networks. A novel cost WSNs are currently exploited in several applications, includ- function is formulated which inherently copes with missing mea- ing environmental monitoring, failure detection or reporting in sures and corrupted data. In particular, the proposed algorithm smart buildings, and target tracking. These applications rely on robustly completes the range map between pairs of sensors by cast- accurate position information of sensor nodes: it is necessary to ing the problem as a low-rank and sparse matrix decomposition, orient the nodes with respect to a global coordinate system in order while constraining the solution to be close to an Euclidean Dis- to report data that is geographically meaningful and to express tance Matrix. Numerical accuracy and computational eciency are all the measurements in a common reference frame. Furthermore, demonstrated by synthetic experiments. The empirical results also basic network services such as routing often rely on location in- show that our method outperforms state-of-the art algorithms in formation. These facts motivate the research on the problem of several scenarios. Sensor Network Localization (SNL), in which sensors self-organize a coordinate system. CCS CONCEPTS Localization methods can provide a global reference frame, meaning that sensor nodes are aligned with some external system, or • Networks → Network algorithms; • Theory of computa- a relative one, meaning that their positions are dened up to an tion → Theory and algorithms for application domains; arbitrary rigid transformation from the global frame. At least three non-collinear anchor nodes are required to dene a global reference KEYWORDS frame in two dimensions, and at least four non-coplanar in tree Robust Matrix Completion, Low-rank and Sparse Decomposition, dimensions. Anchor nodes are ordinary sensor nodes that have a Euclidean Distance Matrix, Euclidean Distance Matrix Completion, priori knowledge about their coordinates. Such information could Sensor Network Localization, Wireless Sensor Networks be hard coded, or acquired through some additional hardware like ACM Reference format: a GPS receiver. Beatrice Rossi, Marco Patanè, Pasqualina Fragneto and Andrea Fusiello. Typically, SNL methods determine the positions of sensor nodes 2017. Robust Localization in Wireless Sensor Networks via Low-rank and either starting from pairwise range or bearing measurements. This Sparse Matrix Decomposition. In Proceedings of ICFNDS ’17, Cambridge, work focuses on the case of range measurements. United Kingdom, July 19-20, 2017, 10 pages. Distances between pairs of nodes can be measured using the DOI: 10.1145/3102304.3102315 Received Signal Strength Indicator (RSSI) of radio signals sent by neighbor sensors. Indeed, the energy of a radio signal diminishes 1 INTRODUCTION with the distance from the signal’s source. As a result, a node listening to a radio transmission should be able to use the strength Wireless Sensor Networks (WSNs) are a collection of autonomous of the received signal to estimate its distance from the transmitter. microelectronic devices distributed over a geographical area (in- Likewise, sensors nodes can asses the Time of Arrival (TOA) or door or outdoor) that cooperate to monitor several physical or the Time Dierence of Arrival (TDOA) of radio or acoustic signals environmental data. Typically, each sensor consists of a low power emitted by neighbor sensors, which in turn can be converted into processor, a limited amount of memory, a sensor board and a wire- pairwise distance estimates. less network transceiver. They constitute an instance of a wireless Measuring distances using RSSI or ToA is subject to noise, rogue measures (or outliers), and missing data. Noise depends on radio Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed propagation which tends to be highly non-uniform, outliers are for prot or commercial advantage and that copies bear this notice and the full citation due to environmental factors (sensitivity to reections, and inter- on the rst page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, ferences) or hardware malfunctioning (suering from transmitter, to post on servers or to redistribute to lists, requires prior specic permission and/or a receiver, and antenna variability), and missing data depend on fee. Request permissions from [email protected]. shields, or limitations in the radio range. ICFNDS ’17, Cambridge, United Kingdom © 2017 ACM. 978-1-4503-4844-7/17/07...$15.00 Our solution to the SNL problem is inspired by recent advances DOI: 10.1145/3102304.3102315 in the eld of Low-Rank and Sparse (LRS) matrix decompositions. ICFNDS ’17, July 19-20, 2017, Cambridge, United Kingdom B. Rossi et al.

We exploit this framework to complete the range information between pairs of sensors in the presence of rogue measures, namely to solve a robust Matrix Completion (MC) problem. In particular, we devise a novel cost function that not only includes unspecied measurements and outliers in its denition, but also constrains the solution to stay close to an Euclidean Distance Matrix (EDM), i.e., a matrix containing squared distances between sensors. We also develop a minimization procedure – dubbed Euclidean R-GoDec, or ER-GoDec for short – that is able to eectively minimize such cost function. Moreover we propose a simple variation of ER-GoDec which eciently handles mobile SNL. Figure 1: The SNL problem. On the left, the input data ma- 2 The paper is organized as follows. Existing solutions to the SNL trix D: colored entries represent distance measurements dij , problem are reviewed in Section 2. Section 3 is an overview of the blackD entries stand for outliers and question marks (?) forD theoretical background on Robust Matrix Completion via LRS ma- unspecied measurements. On the right, the computed po- trix decomposition and EDM Completion. In Section 4 we describe sitions of the sensor nodes. our method ER-GoDec and its variation for eciently addressing mobile networks. In Section 5 we show the results of several experiments on synthetic data and we highlight the benets introduced by We now review the most common localization techniques which applying our method. The conclusions along with possible further appeared throughout the literature. ˆ developments are presented in Section 6. If all pairwise distance measurements are available, namely D is complete, the simplest approach to solving the SNL problem is the 2 KNOWN SOLUTIONS AND LIMITATIONS classical Multidimensional Scaling (MDS) [8], originally proposed by [29]. MDS exploits the following relationship between the Gram We rst introduce some notations and basic concepts about SNL. matrix G = XXT and the associated EDM matrix Suppose we have n sensors which we refer to as the nodes of the network. We denote the positions of the nodes by a set of n points 1 G = − JDJ := T (D) (2) in Rm (in our case m = 2, 3) ascribed to the rows of a matrix 2 ∈ Rn×m T 1 T X , X = [x1, x2,..., xn] . The distance between node i where J is the centering operator dened as J = I − n 11 (1 denotes and j is dij = kxi − xj k2, and the associated Euclidean Distance the column vector of all ones and I the identity matrix). In particular, Matrix D ∈ Rn×n is dened as the matrix of squared distances MDS minimizes the following cost function between the nodes, i.e. minkT (D) − XXT k2 . (3) X F 2 2 m Dij := dij = ||xi − xj ||2 . (1) The point coordinates X ∈ RD can be found by computing the eigen-decomposition of T (D) truncated at the m largest eigen- In real world applications, we rarely have a perfect EDM. We values. MDS is fast and closed-form, however it is ineectual in D know that the measured distances could be aected by noise, out- practice as it does not manage unspecied measurements, and it is ˆ liers, and often we can acquire just a subset. We denote as D the not robust to outliers. MDS is summarized in Algorithm 1. matrix of squared measured distances, henceforth called data ma- Observe that applying MDS is straightforward, thus there is no trix. practical dierence in localization methods between recovering The goal of localization methods is to recover the point coordi- X, or rather the Gram Matrix G, or the EDM D. Indeed, point ˆ nates X from D. An illustration of the SNL problem is shown in coordinates can always be derived from G or D by carrying out Figure 1. MDS (in whole or in part). In range-based SNL, it is useful to model the sensor network using the distance graph G = (V, E), where vertices represent MDS the sensor nodes, and edges correspond to the pairs of sensors for Algorithm 1 which a distance measurement is available. SNL can also be seen Require: D, m, as a graph realization problem, where the goal is to assign to each Ensure: X, vertex coordinates x ∈ Rm so that the Euclidean distances between D1 T i J ← I − n 11 pairs of nodes match the distances of the corresponding edges. 1 G ← − 2 JDJ = T (D) A sensor network is uniquely localizable if exists a unique X [Q, Λ] ← eig(G) that realizes the distances in D (up to a rigid transformation). It is D1/2 D X ← Qm Λm demonstrated in [11] that if the distance graph is generically globally D rigid – GGR for short –, then the network is uniquely localizable. Jackson et al. [22] proved that in two dimensions the distance graph In case of unspecied measurements, the data matrix is a partial is GGR if and only if it satises two properties that can be veried matrix. This means that the entries of D are specied on a sampling in polynomial time, namely 3-connectivity and redundant rigidity. set, namely a subset of index pairs, and unspecied elsewhere. Let D For more details about these concepts see [22]. Ω be a (0, 1)-matrix representing the sampling set of D, i.e. Ωij = 1 D Robust Localization in Wireless Sensor Networks ICFNDS ’17, July 19-20, 2017, Cambridge, United Kingdom if Dij is specied and 0 otherwise, then we dene PΩ (D) = Ω Conventional solvers for MC, as for example SVT [10], ALM [25], D = A D, where is the Hadamard (entry-wise) product (where APGL [28] and OptSpace [23], can ll unspecied entries, but they D D unspecied value multiplied by 0 gives 0) and A is the adjacency are not robust to the presence of outlier measurements in the data D D matrix of the distance graph G. matrix. An approach to solving the SNL problem in case of missing data In order to handle outlier measurements, one possibility is to is by minimizing the following stress function cast MC as a Low-Rank and Sparse matrix decomposition problem. LRS decompositions address the general problem of decomposing 1/2 1/2 2 a data matrix as the sum of a low-rank term representing some minkPΩ (D − D )kF . (4) X meaningful low-dimensional structure contained into the data, a One iterative method to solveD (4) is a variant of the gradient de- sparse term representing rogue measures and a term accounting for scent called stress majorization algorithm, also known as SMACOF a diuse noise. Such a problem arises in a number of applications [8, 18], originally proposed by [13]. SMACOF is computationally in Computer Vision, as for example in the separation of foreground ecient, however, it is not robust to outliers. objects from the background [3], or in synchronization problems The state-of-the-art approach to solve localization is based on in Structure from Motion [1, 2]. A wide overview of LRS matrix Semidenite Programming (SDP) [6, 7]. SDP is robust and copes decompositions and their applications can be found in [9]. with unspecied measurements. The idea is to consider the follow- Let us consider the following LRS decomposition problem ing cost function

PΩ (D) = PΩ (L) + S + N . (8) minkPΩ (D − D)k1 (5) X Here PΩ (D) is the partialD data matrix, L is an unknown low-rank which can be rewritten in terms ofD the Gram matrix as term representing a cleaned completion of the data matrix, S is an D unknown sparse matrix representing outliers, and N is a diuse T minkPΩ (D − K (G))k1 s.t. G = XX (6) G noise. The problem of computing L (and eventually S) from PΩ (D) K is known as Robust Matrix Completion (RMC). RMC is illustrated where (G) is denedD as D in Figure 2. K (G) := diag(G)1T − 2G + 1diag(G)T = D. (7) Observe that this is another form of the relationship between the Gram Matrix and the associated EDM. The minimization in (6) is then cast into a Semidenite Programming by relaxing the constraint G = XXT to G 0, and the resulting problem is solved by interior point methods (as explained in [20]). Naturally, since the minimization variable is G, a partial MDS is then needed to recover the point coordinates X ∈ Rm. SDP provides very accurate results. However it does not scale well and relies on sophisticated optimization procedures that cannot be easily implemented on devices with limited resources. After running any localization procedure which nds point co- Figure 2: Robust Matrix Completion. On the left, the input ordinates X, the obtained solution can be expressed in a global matrix: Colors represent the low-rank structure, black en- coordinate frame using some xed anchor nodes (at least 3 or 4 in tries represent outliers and symbols ? represent unspecied our case) exploiting a rigid registration routine. A common proce- entries. On the right, the completed (and cleaned) low-rank dure exploits Orthogonal Procrustes Analysis [17]. matrix. 3 MATHEMATICAL BACKGROUND Our solution to the SNL problem exploits a Low-rank and Sparse R-GoDec [1] is an algorithm that solves RMC under the LRS matrix decomposition framework. R-GoDec is built on GoDec decomposition to robustly complete PΩ (D). In particular, our [30] which was originally designed to solve the conventional MC method constrains the completion of the partial matrix P (D) to D Ω problem only. R-GoDec splits the sparse term as the contribution approach a Euclidean Distance Matrix, namely attempts to compute D of two terms S and S with complementary supports: a EDM completion of the data matrix. For these reasons, we now 1 2 briey review the main concepts about robust Matrix Completion • S1 is a sparse matrix with support on Ω representing outlier via LRS decompositions, and EDM completion. measurements; • S2 has support on f (where f is the complement of Ω), 3.1 Robust Matrix Completion via LRS Matrix and it is an approximation of −Pf(L), representing the Decompositions completion of unspecied entries. Matrix Completion is concerned with the problem of recovering This results in the following decomposition problem unspecied entries of a low-rank partial matrix. A completion of a partial matrix consists in ascribing values to the missing data. PΩ (D) = L + S1 + S2 + N (9) D ICFNDS ’17, July 19-20, 2017, Cambridge, United Kingdom B. Rossi et al.

which is translated into the minimization approaches to innity. As a consequence, the method can be slow, 1 2 but it can be useful if we have some ecient method, such as closed- min PΩ (D) − L − S1 − S2 + λ kS1 k1 L,S1,S2 2 F (10) form, for the projections. n s.t. rank( L) ≤ rD, supp(S1) ⊆ Ω , supp(S2) = f Exploiting the geometric characterization of EDM , and the  closed-form projections on K and K , APA can be applied to R-GoDec solves the problem above using a block-coordinate 1 2 get an EDM which is close to the output of a RMC method. This scheme that alternates the L, S , and S updates. L is updated as the 1 2 methodology is depicted in Figure 3. rank-r projection of P (D) − S − S using Bilateral Random Pro- Ω 1 2 Mishra et. al. in [26] proposed a Manifold Based-Optimization jections (BRP) [19]. S is updated by applying the soft-thresholding 1 D method – MBO for short – by casting the EDM completion into Θ P − −P operator λ [5] to the matrix Ω (D L). S2 is updated as f(L). a minimization over the set of low-rank positive semidenite ma- These steps are iterated until convergence. D trices. In particular, they derived an unconstrained optimization problem on a smooth Riemannian manifold which they tackled via 3.2 Euclidean Distance Matrix Completion gradient descent. n×n A matrix D ∈ R is an EDM if and only if there exist n Another possibility is to reformulate the EDM completion as m n×m points in R ascribed to the row of a matrix X ∈ R , X = an instance of the SDP, as explained in [16, 24]. Indeed, since a T [x1, x2,..., xn] , such that the squared distance between xi and symmetric zero-diagonal matrix M is an EDM if and only if T (M) 2 xj is given by Dij = kxi − xj k2 . In particular, every EDM is a 0, then forcing T (M) to be PSD is the same of imposing M to belong non-negative symmetric matrix with zero diagonal. to K2. Thus, intuitively, the SDP method, while searching for the A notable fact about EDMs is that their rank is at most m + 2 Gram Matrix G, also nds an EDM completion of PΩ (D). (this can be easily derived from (7)). As for the SNL problem, the rank of the associated EDM matrix is thus at most 4 (if m = 2) D or 5 (if m = 3). This result also states that the rank of an EDM is independent of the number n of points. Thus, since we can suppose that m << n in the SNL problem, the associated EDM is low-rank. Other notable facts about EDMs regard their geometrical structure. The set of all EDMs in Rn×n forms a closed convex cone n referred to as EDM . Furthermore, let K1 and K2 be two convex cones dened as follows

K n 1 = Sh (11) n K2 = {M ∈ S s.t. − JMJ 0} (12) Sn × Sn where is the set of n n symmetric matrices, h is the set of n × n symmetric matrices with zero diagonal and J is the centering operator. Then it is demonstrated in [12] that EDMn is the intersection of K1 and K2. There also exist analytic formulas to project n n Figure 3: Alternating projections algorithm on EDM . symmetric matrices respectively on K1 and K2. Let be M ∈ S , then the projections are given by

P − 4 PROPOSED METHOD K1 (M) = M diag(M) (13) P (M) = M − P n (JMJ ) (14) In this section we describe an algorithm we dubbed ER-GoDec, K2 S+ + T T that robustly solves the EDM completion problem. where P n (JM J ) := QΛ Q , QΛQ is the eigen-decomposition S+ Starting from the LRS decomposition proposed by R-GoDec in of JM J with eigenvalues arranged in non-increasing order, and + Eq. (9), we constrain the completion of PΩ (D) to belong to K1 and Λ = max{0, Λii } for i = 1,...,n. ii K2 through the following minimization problem Let now PΩ (D) be a partial EDM. A matrix D is an EDM com- D 2 pletion if D is a completion of PΩ (D) and it is an EDM. In other 1 D min PΩ (D) − L − S1 − S2 + λ kS1 k1 terms, the EDM completion is the problem of nding an EDM that L,S1,S2 2 F D completes a given partial EDM. s.t. rank( L) ≤ mD + 2,  We briey recall some of the available approaches to solve the L ∈ K ,  1 (15) EDM completion. Observe that, in general, algorithms for Robust  L ∈ K2, Matrix Completion do not guarantee that the completion of a partial  supp(S ) ⊆ Ω, EDM is again an EDM.  1  A rst possibility is to combine conventional solvers for RMC supp(S2) = f.  with the Alternating Projections Algorithm (APA) [14]. Given two This problem can be solved using the procedure reported in convex sets A, B and a starting point M, APA alternately projects Algorithm 2. As in R-GoDec, variables L, S1 and S2 are updated 2 the point on A and B in order to reach a point in their intersection. iteratively one at a time. A rank-(m + 2) projection of PΩ (D ) − In [4], it is shown that APA converges as the number of iteration S − S is performed in order to nd the low-rank term L. The 1 2 D Robust Localization in Wireless Sensor Networks ICFNDS ’17, July 19-20, 2017, Cambridge, United Kingdom projection is computed using Bilateral Random Projections (BRP) Up to this point we worked under the same assumption, albeit [19]. Then, the resulting L is rst projected on K2 and then on K1 not necessary. As a matter of fact, ER-GoDec, can easily cater for according to Eq. (14) and Eq. (13) respectively. Finally, S1 is updated mobile sensors. We now present a simple variation of the ER-GoDec via soft-thresholding of PΩ (D − L) and −PΩ (L) is assigned to S2. algorithm, dubbed Incremental ER-GoDec or IER-GoDec, which eciently addresses mobile SNL, namely networks where a subset D Algorithm 2 ER-GoDec of sensors is moving. We propose to rst perform localization as explained in Section Require: D, Ω, m, ϵ, λ 4 using ER-GoDec. Then, localization in subsequent time points Ensure: L, S , S D 1 2 can be obtained using IER-GoDec. If consecutive network con- Initialize: L = D, S1 = 0, S2 = 0 2 2 gurations are suciently close one to each other, then applying P (D) − L − S − S / P (D) > ϵ while Ω D 1 2 F Ω F do IER-GoDec solves the SNL problem more quickly than applying (1) L ← rank-(m + 2) approximation of P (D) −S −S ER-GoDec afresh with new range measurements. D D Ω 1 2 via BRP IER-GoDec exploits an alternative initialization of the variables ← P D with respect to ER-GoDec. Let us suppose that at time τ = t the (2) L K2 (L) ← P localization is performed via ER-GoDec, and that St and St are (3) L K1 (L) 1 2 t+1 (4) S1 ← Θλ (PΩ (D − L)) the sparse terms computed by the algorithm. Let PΩ (D ) be the (5) S ← −P (L) partial data matrix at time at time τ = t + 1. Then IER-GoDec 2 f D D t+1 − t − t end while initializes the low-rank term L as D S1 S2 and then continues implementing the same steps as ER-GoDec. D Once the EDM completion problem is solved via ER-GoDec, the Again, once the EDM completion problem is solved, the point t+1 point coordinates X can be obtained by applying the MDS algorithm coordinates X can be obtained by applying the MDS algorithm t+1 to L. The complete procedure is depicted in Figure 4. to L . The Incremental ER-GoDec is summarized in Algorithm 3.

Algorithm 3 IER-GoDec t+1 t t Require: D , S1, S2, Ω, m, ϵ, λ L, St+1, St+1 Ensure: D 1 2 t+1 − t − t t+1 t t+1 t Initialize: L = D S1 S2, S1 = S1, S2 = S2 2 2 P (Dt+1) − L − St+1 − St+1 / P (Dt+1) > ϵ while Ω D 1 2 F Ω F do (1) L ← rank-(m + 2) approximation of P (Dt+1) − D D Ω t+1 − t+1 S1 S2 via BRP ← P D (2) L K2 (L) ← P (3) L K1 (L) t+1 ← Θ P t+1 − (4) S1 λ ( Ω (D L)) (5) St+1 ← −P (L) 2 f D end while

5 EXPERIMENTS In this section we validate our solution through synthetic experiments in MATLAB®. We rst analyze some algorithmic features of ER-GoDec. In par- Figure 4: SNL as a robust EDM completion problem. On the ticular, we tune the soft-thresholding parameter λ and we evaluate top left, the input data matrix: colors represent distance some alternative iteration schemes. Then, we compare ER-GoDec measurements dij , black entries represent outliers and sym- respectively with two methods for SNL (SDP and SMACOF), a com- bols ? represent unspecied entries. On the top right, the D bination of a solver for RMC and APA (R-GoDec + APA), and an completed (and cleaned) EDM. On the bottom, the positions algorithm for EDM completion (MBO). The implementations of of the sensor nodes computed applying the MDS algorithm SDP [6] (based on the SDPT3 solver [27]), MBO [26] and SMACOF to the EDM. are available online, while for ER-GoDec and R-GoDec + APA we use our implementations. We compare the performances of the methods in terms of noise resilience, robustness to outliers, sensi- 4.1 Incremental ER-GoDec for mobile SNL tivity to missing data, and eciency. Finally, we assess IER-GoDec Most of the state-of-the-art techniques assume that sensor nodes in a mobile scenario. are in xed positions, and if they move a new instance of SNL has In each experiment, we generate n random points in a squared to be solved from scratch. box of side l centered in the origin, representing the ground truth ICFNDS ’17, July 19-20, 2017, Cambridge, United Kingdom B. Rossi et al. node positions. We perturb the true distances between points with outer iterations, with the purpose of attaining best eciency and Gaussian noise with 0-mean and standard deviation σ. We discard accuracy. Besides the ER-GoDec algorithm described in Algorithm a portion (H%) of distances to simulate unspecied data. We set 2, we analyze two dierent iteration schemes. The rst scheme a portion (O%) of the specied distances to random values in the projects L alternatively on K2 and K1 5 times for each outer iter- interval [0,l] to simulate outliers. We check the localizability of the ation. The second scheme projects L on K2 and K1 every 5 outer distance graph using our implementations of redundant rigidity iterations. We denote the two schemes respectively as ER-GoDec-A [21] and 3-connectivity [15]. and ER-GoDec-B. The error measure we used is the Root Mean Square Error (RMSE) In Figure 6 we plot the residuals corresponding to ER-GoDec and r the two schemes just dened as function of the number of outer 1 2 RMSE = X − X˜ (16) iterations. For each scheme, the residue is dened as kPΩ (D) − L − n F S − S k2 /kP (D)k2 . The considered parameter conguration is 1 2 F Ω F D where X˜ are the estimated point coordinates and X the true ones. n = 100, l = 100, H = 50%, O = 20%, and σ = 0.6. The RMSE is computed after the alignment of X˜ with respect to X˜ D obtained using Orthogonal Procrustes Analysis [17] (also available in MATLAB® through the procrustes command). In each experiment, results are averaged over 100 dierent simulations minus the number of tests in which the localizability check fails.

5.1 ER-GoDec Setup We rst study the behavior of ER-GoDec with respect to the soft-thresholding parameter λ. Figure 5 shows the (average) RMSE obtained using ER-GoDec with dierent values of λ. The considered parameter conguration is n = 100, l = 100, H = 50%, O = 20%, and σ = 0.6.

Figure 6: Behaviors of dierent iteration schemes. On the 15 x-axis the number of iterations, on the y-axis the residuals. The number of sensors is n = 100, the box side is l = 100, the portion of unspecied entries is H = 50%, the portion of 10 outliers is O = 20% and the noise std deviation is σ = 0.6. RMSE The resulting RMSE values are respectively 0.383 for ER-GoDec, 5 0.379 for ER-GoDec-A, 0.402 for ER-GoDec-B. Figure 6 shows that ER-GoDec and ER-GoDec-A are almost equivalent both in terms of convergence speed (measured in number of outer iterations) 0 and accuracy (measured by RMSE). However, ER-GoDec-A is less 0 100 200 300 400 500 ecient than ER-GoDec since each outer iteration of ER-GoDec- A is computationally about 5 time more complex than an outer iteration of ER-GoDec. Conversely, the residuals of ER-GoDec-B Figure 5: Performances of ER-GoDec as a function of λ. On present an undesired oscillating behavior, and the method stops the x-axis the values of λ, on the y-axis the corresponding after reaching the maximum number of iterations (1500). On the (average) RMSE obtained using ER-GoDec. The number of whole, we choose ER-GoDec as reference method, since it balances sensors is n = 100, the box side is l = 100, the portion of the trade-o between accuracy and eciency. unspecied entries is H = 50%, the portion of outliers is O = 20% and the noise std deviation is σ = 0.6. 5.2 Noise Resilience In this experiment we evaluate the noise resilience of ER-GoDec ER-GoDec achieves the best performances (RMSE ≤ 0.5) for without considering unspecied and outliers data. In particular we λ ∈ [25, 150], hence a reasonable value for λ can be picked within set n = 100, l = 100, H = 0% (no undened entries), and O = 0% (no this interval. Since similar trends have been observed also with outliers). Clearly in this case the localizability test is trivial, since other parameter congurations, we choose to x the value λ = 50 the input matrices are complete. Results are shown in Figure 7. for all the next experiments. Observe that SMACOF achieves the most accurate results, fol- We are also interested in tuning the number of internal projec- lowed by MBO. This depends on the fact that the SMACOF maxi- tions of ER-GoDec (respectively on K2 and K1) with respect to mizes a likelihood function (stress), hence it is statistically optimal Robust Localization in Wireless Sensor Networks ICFNDS ’17, July 19-20, 2017, Cambridge, United Kingdom

O = 0%

ER-GoDec ER-GoDec 0.3 SMACOF 40 SMACOF SDP SDP 0.25 R-GoDec + APA R-GoDec + APA MBO 30 MBO 0.2

RMSE 20

0.15 RMSE

0.1 10 0.05

0 0 0 0.2 0.4 0.6 0.8 1 50 60 70 80 90 noise ( ) unspecified entries (H%) O = 30% Figure 7: Noise Resilience. On the x-axis the noise std deviation σ, on the y-axis the corresponding (average) RMSE ER-GoDec 40 SMACOF obtained with several methods. The number of sensors is SDP n = 100, the box side is l = 100, the portion of unspecied R-GoDec + APA entries is H = 0% and the portion of outliers is O = 0%. 30 MBO

20 RMSE in presence of i.i.d. Gaussian noise. The robust methods, namely ER-GoDec, R-GoDec + APA and SDP, are sub-optimal because they 10 trade statistical eciency for robustness. All errors grow linearly with respect to the noise std deviation. Thus we x a reasonable level of noise, namely σ = 0.6, for the following experiments. This 0 level of noise provides an RMSE approximately lower than 0.2 for 50 60 70 80 90 all the methods. unspecified entries (H%)

5.3 Sensitivity to Unspecied Data Figure 8: Sensitivity to unspecied data. On the x-axis the portion of unspecied entries H%, on the y-axis the (aver- In this experiment we study the sensitivity of ER-GoDec to age) RMSE obtained with several methods. The number of . unspecied data. We set n = 100, l = 100, σ = 0 6 and two portions sensors is n = 100, the box side is l = 100 and the noise std of outliers, namely O = 0% (no outliers) and O = 30%. Results are deviation is σ = 0.6. The portions of outliers are respectively shown in Figure 8. O = 0% (no outliers) in the upper plot and O = 30% in the In the case without outliers (upper plot of Figure 8), all the lower plot. methods are insensitive to unspecied entries up to a portion of the 50%. When the portion of unspecied entries is between the 50% and the 80%, ER-GoDec achieves similar results compared to SDP, better results compared to R-GoDec + APA and MBO, while SMACOF In Figure 9 we x a set of parameters in order to appreciate the performances get worse. When the portion of unspecied entries results of a single simulation. As expected, ER-GoDec provides is greater than the 80%, the accuracy drops for all the methods, the most accurate localization, followed by SDP and R-GoDec + although ER-GoDec performances still remain acceptable. APA which fail in localizing a couple of sensors (located near the The lower plot of Figure 8 shows that SMACOF and MBO become top-right border). Instead SMACOF and MBO obtain inaccurate sensitive to unspecied entries when outliers are present (although positions for almost all the sensors. The bottom-right plot is the ˜ not reported in the gure, their RMSE stands at around 10 even for scatter plot between D(X ) (the ground truth EDM) and D(X )) (the portions of unspecied entries lower than or equal to the 50%). R- distance matrix associated to the computed positions). Observe that GoDec + APA, ER-GoDec and SDP are not sensitive to unspecied distance values computed by of SDP are almost undetectable to entries up to a portion of the 50%, even in the presence of outliers. those computed by ER-GoDec. When the percentage of unspecied entries is greater than the 50% all the methods worsen. ER-GoDec shows the best breaking point 5.4 Robustness to Outliers at the 70%. In this experiment we analyze the robustness to outliers of ER- In our 100 simulations the localizability check fails approximately GoDec. We set n = 100, l = 100, σ = 0.6 and two portions of 20 times when the portion of unspecied entries is greater than the unspecied entries, namely H = 0% (no unspecied entries) and 80%, while for portions lower than the 80% the check never fails. H = 60%. ICFNDS ’17, July 19-20, 2017, Cambridge, United Kingdom B. Rossi et al.

SMACOF RMSE = 5.67 MBO RMSE = 5.07 50 50 H = 0% 40 40 30 ER-GoDec 30 30 SMACOF 20 20 25 10 10 SDP 0 0 R-GoDec + APA -10 -10 20 MBO -20 -20 -30 -30 15

-40 -40 RMSE -50 -50 10 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50

ER-GoDec RMSE = 0.37 R-GoDec + APA RMSE = 0.86 50 50 5 40 40 30 30 0 20 20 0 10 20 30 40 50 10 10 outliers (O%) 0 0 H = 60% -10 -10 -20 -20 30 ER-GoDec -30 -30 SMACOF -40 -40 25 SDP -50 -50 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 R-GoDec + APA 20 MBO SDP RMSE = 0.57 50 140 ER-GoDec 40 15 120 SMACOF 30 SDP RMSE 100 R-GoDec + APA 20 MBO 10 10 80 0 60 5 -10 -20 40 0 -30 20 0 10 20 30 40 50 -40 0 outliers (O%) -50 0 20 40 60 80 100 120 140 -50 -40 -30 -20 -10 0 10 20 30 40 50 Figure 10: Robustness to outliers. On the x-axis the portion Figure 9: Localization of a network of n = 100 sensors in a of outliers O%, on the y-axis the (average) RMSE. The num- box [−50, 50] (i.e. l = 100) with noise std deviation σ = 0.6. The ber of sensors is n = 100, the box side is l = 100, and the noise portion of unspecied entries is H = 60% and the portion of std deviation is σ = 0.6. The portion of unspecied entries outliers is O = 10%. Green circles represent the true point co- are respectively H = 0% (no unspecied entries) in the upper ordinates, red stars the estimated ones and segments the dis- plot and H = 60% in the lower plot. crepancies between them. The bottom-right plot is the scat- ˜ ter plot between D(X ) and D(X ). As for the localizability check, we observe that with these parameter congurations the check never fails. Results in the upper plot of Figure 10 conrm that ER-GoDec, Figure 11 conrms that the accuracy of the localization is more R-GoDec + APA and SDP are robust, since they obtain rather good inuenced by outliers rather than by random unspecied entries. results up to a portion of outliers of the 40%. On the contrary, Furthermore, only ER-GoDec provides an accurate localization, SMACOF and MBO are non-robust, indeed the RMSE is considerable while R-GoDec + APA and SDP cannot manage such portions of also for portion of outliers lower than the 5%. The lower plot of unspecied data and outliers. SMACOF and MBO get completely Figure 10 shows that ER-GoDec has the best breaking point at inaccurate localizations. The scatter plot between D(X ) and D(X˜ ) the 30%, achieving accurate results also for high portions of both (the bottom-right plot) veries that only ER-GoDec accurately unspecied data and outliers, while R-GoDec + APA and SDP recovers the distance matrix. obtain worse results. SMACOF and MBO produce highly inaccurate positions. 5.5 Execution Time An explanation for these results is that ER-GoDec and R-GoDec In this experiment we evaluate the execution time as a function + APA correctly identify (and discard) outliers because they are of the number of sensors. We consider l = 100, σ = 0.6, H = 30% robust MC methods. The robustness of SDP depends on the fact and O = 10%. Results are reported in Figure 12. that it minimizes a robust cost function (based on `1-norm). On the From the upper plot of Figure 12 we can see that SDP is drastically contrary, SMACOF and MBO fail in identifying outliers since they slow (10 to 50 times slower than the other methods). Also R-GoDec + minimize non-robust cost functions (based on the Frobenius norm). APA is quite slow with respect to SMACOF, ER-GoDec and MBO. In Robust Localization in Wireless Sensor Networks ICFNDS ’17, July 19-20, 2017, Cambridge, United Kingdom

SMACOF RMSE = 10.27 MBO RMSE = 10.88 50 50 40 40 40 ER-GoDec 30 30 SMACOF 20 20 SDP 10 10 30 R-GoDec + APA 0 0 -10 -10 MBO -20 -20 20 -30 -30

-40 -40 time (s) -50 -50 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50 10 ER-GoDec RMSE = 0.58 R-GoDec + APA RMSE = 6.24 50 50 40 40 30 30 0 20 20 10 20 30 40 50 60 70 80 10 10 n 0 0 -10 -10 7 -20 -20 ER-GoDec -30 -30 6 SMACOF -40 -40 MBO -50 -50 5 -50 -40 -30 -20 -10 0 10 20 30 40 50 -50 -40 -30 -20 -10 0 10 20 30 40 50

SDP RMSE = 4.31 4 50 140 ER-GoDec 40 120 SMACOF 3 30 SDP time (s) 100 R-GoDec + APA 20 MBO 2 10 80 0 60 1 -10 -20 40 0 -30 20 50 100 150 200 250 300 -40 0 n -50 0 20 40 60 80 100 120 140 -50 -40 -30 -20 -10 0 10 20 30 40 50 Figure 12: In the upper plot, the execution time of all the Figure 11: Localization of a network of n = 100 sensors in a methods. In the lower plot, execution time of SMACOF, ER- box [−50, 50] (i.e. l = 100) with noise std deviation σ = 0.6. GoDec and MBO. On the x-axis the number of sensor nodes The portion of unspecied entries is H = 60% and the por- (n), on the y-axis the execution time (in seconds). The box tion of outliers is O = 30%. Green circles represent the true side is l = 100, the portion of unspecied entries is H = 30%, point coordinates, red stars the estimated ones and segments the portion of outliers is O = 10% and the noise std deviation the the discrepancies between them. The bottom-right plot is σ = 0.6. is the scatter plot between D(X ) and D(X˜ ).

τ = t + 1 from scratch. The accuracy of the two methods is the order to compare the execution time of the most ecient methods, same. we increased the number of sensors excluding SDP and R-GoDec + We can appreciate that the speed-up is signicantly high if few APA from the evaluation. The lower plot of Figure 12 shows that sensors are allowed to move. Anyway, even if all sensors are moving MBO and SMACOF are noticeably faster with respect to ER-GoDec, the speed up is still favorable to IER-GoDec. even if ER-GoDec still remains acceptable. We can conclude that IER-GoDec is a promising approach to We observe that with this portion of unspecied entries (30%) solve the SNL problem in mobile scenarios as those described in the localizability check never fails. Section 4.1.

5.6 IER-GoDec 6 CONCLUSIONS We evaluate Incremental ER-GoDec through the following ex- In this paper we have presented ER-GoDec: a novel algorithm periment: we proceed as explained in Section 5 for generating Dt which exploits a Low-Rank and Sparse matrix decomposition tai- and we localize the network using ER-GoDec. Then we move ap- lored to SNL. We argue that the benet yielded by ER-GoDec is D proximatively one unity (in the plane) a random subset of sensors twofold. First, it is both ecient and highly accurate, as demon- generating Dt+1. Figure 13 shows the speed-up obtained by using strated by synthetic experiments. Such occurrence permits to lo- IER-GoDec at time τ = t + 1 rather than using ER-GoDec at time calize the network using low-power processor. The second benet D ICFNDS ’17, July 19-20, 2017, Cambridge, United Kingdom B. Rossi et al.

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