Sparsity-Based Direct Location Estimation Based on Two-step Dictionary Learning

This paper proposes an adaptive sparsity-based direct position 
determination (DPD) appoach to locate multiple targets in the case of 
time-varying channels. The novel feature of this method is to dynamically 
adjust both the overcomplete basis and the sparse solution based on a two-step 
dictionary learning (DL) framework. The method first performs supervised 
offline DL by using the quadratic programming approach, and then the dictionary 
is continuously updated in an incremental fashion to adapt to the time-varying 
channel during the online stage. Furthermore, the method does not need the 
number of emitters a prior. Simulation results demonstrate 
the performance of the proposed algorithm on the location estimation accuracy.


Introduc
arsity-B 1 The Na 2 The Jiangsu 3  pose an adaptive sparsity-based DPD (ASDPD) algorithm to dynamically adjust both the overcomplete basis and the sparse solution so that the solution can better match the actual scenario.The method first performs supervised offline dictionary training by using the quadratic programming approach.During the online stage, the dictionary is continuously updated in an incremental fashion to adapt to time-varying factors.
The notation used in this paper is according to the convention.Symbols for matrices (upper case) and vectors (lower case) are in boldface.The remainder of the paper is organized as follows.Section 2 briefly describes the system model assumed throughout this paper and formulates as a sparse recovery problem.In Section 3, we introduce a scheme calibrating the overcomplete basis dynamically and estimating the sparse solution adaptively.Simulation results are given in Section 4. Finally, Section 5 concludes the paper.

System Model and Problem Formulation
Consider N base stations (BS) intercepting the narrowband signals transmitted by L possible sources.Each BS which knows its coordinates is equipped with an antenna array consisting of M elements.Denote the lth unknown target position by the vector of coordinates l P .We use the far-field point-target model, which is commonly used for source localization due to its simplicity [3,4,9].Based on this model, the received signal observed by the nth BS is given by 1 where ( ) a p is the array response at the nth BS from a signal transmitted position, and the propagation delay from the lth transmitter to the nth BS is given by ( ) represents noise terms, which is assumed as the independent and identically distributed (i.i.d.) complex Gaussian process, uncorrelated with the signals.
We divide the area of interest into K grids.In general, . Then, we formulate the location problem as a following CS problem , {1, , } , , ] is an overcomplete basis matrix at the nth BS, and ( ) corresponds to the noiseless signal vector between the ith grid and the nth BS.
, we can obtain Note that H is known under the ideal channel condition, which means that we can estimate the actual coordinates of targets as long as we find the positions of nonzero values in θ .That is, the problem of localization is converted into one of sparse signal recovery from (3).Moreover, the number of these dominant nonzero values gives L.
However, the non-ideal factors are inevitable in a practical localization system.These factors include the channel attenuation, phase error, time-varying fluctuations of the radio channel and so forth.When these happen, the predefined dictionary cannot effectively express the actual signal, which will cause performance degradation in sparse recovery process.
For avoiding the difficulty of estimate all kinds of the time-varying factors, we assume the error dictionary matrix Γ which describe the difference between the predefined dictionary and the practical received signals.Note that the error matrix Γ is time-varying and cannot be known in advance.In this scenario, the sparse positioning model is correspondingly modified as: where = D ΓH denotes the actual overcomplete basis with the time-varying interference.To prevent D from having arbitrarily large values (which would lead to arbitrarily small values of θ ), it is common to constrain its columns to have a 2 l norm less than or equal to one.Obviously, the mismatch exists between the columns of D and the corresponding columns of the predefined basis H , and thus the performance degradation is inevitable in the sparse recovery process.Focused on this problem, an adaptive sparse recovery algorithm is proposed in this paper, which dynamically calibrate the overcomplete basis so that the sparse solution can better fit the actual scenario.

Sparse Representation Based on the Two-stage Dictionary Learning
The key feature of adaptive sparse recovery is the adap-tive adjustment of the overcomplete basis.This process generally learns the uncertainty of the dictionary, which is not available from the prior knowledge, but rather has to be estimated using a given set of training samples.Several different DL algorithms have been presented recently [10].However, these methods generally cannot effectively handle very large training sets or dynamic training data changing over time.To overcome these shortcomings, we propose a two-stage DL approach that can adapt to the varied upcoming samples.So far, the most DL methods are generally based on alternating minimization.In one step, a sparse recovery algorithm finds sparse representations of the training samples with a fixed dictionary.In the other step, the dictionary is updated to decrease the average approximation error while the sparse coefficients remain fixed.The proposed method in this paper also uses this formulation of alternating minimization.

Sparse Recovery Phase
The above problem of noisy sparse signal recovery can then be converted into a following optimization problem where λ is the regularization parameter.However, it should be emphasized that larger coefficients in θ are penalized more heavily in the 1 l norm than smaller coefficients, unlike the more democratic penalization of the 0 l norm [11].In practice, large coefficients are usually the entries corresponding to the actual positions of targets, while small coefficients commonly represent the noise entries.The imbalance of the 1 l norm penalty will seriously influence the recovery accuracy, which may result in many false targets.Therefore, in this paper we choose the reweighted 1 l norm minimization algorithm in [11] as our sparse recovery method, which can overcome the mismatch between 0 l norm minimization and 1 l norm minimization while keeping the problem solvable with convex estimation tools.

Dictionary Learning Phase
In this paper, we propose a two-stage DL framework in which the offline DL method allows to train the dictionary in a supervised manner to integrate the large training sets, and the incremental DL method based on the results in the offline stage handles the unseen online variation to enhance its adaptability.1) Offline dictionary learning In this stage, the ideal overcomplete basis H is optimized to better represent the data of the training sets.Since the sparse coefficients θ are fixed in the DL stage, the resulting optimization problem becomes: Let's introduce several new expressions for clarity of notation vec( ) vec( ) Omitting the terms that do not depend on D, the objective function in ( 6) can be equivalent to Note that ( 8) is a standard form of constrained quadratic programming problem which can be solved by any standard optimization method, such as the gradient projection algorithm in [12].Moreover, the matrix G is obviously a positively definite matrix, and thus ( 8) is convex function and can be guaranteed to find a global optimum [13] in this DL phase.
2) Online dictionary learning Although the offline DL stage has adjust the overcomplete basis according the training data, it is impossible to be fit for all kinds of time-varying interference patterns.Moreover, its computation load is quite large for realtime localization.On the contrary, the online incremental learning is especially applicable when one seeks to find the variation in the sense that the time-varying channels pattern might not be specifically learned offline but can be distinguished from the past online observations.Based on the incremental learning pattern, the online learning algorithm in [14] can use the result of the offline DL stage as a warm restart for computing the next dictionary where the new samples will be fed into the online dictionary learning procedure, and thus a single iteration has empirically been found to be enough [14].
For completeness, a full description of the algorithm is given in Algorithm 1. 1) update the pre-trained dictionary according to the latest observations using the online DL algorithm in [14] with off D as warm restart, and return the learned dictionary (1)   on D ; 2) use the reweighted 1-norm algorithm to minimize the objective function in (6) with respect to (1)   on θ keeping (1)   on D fixed; 3) (0) (1) on on ˆ= D D ; (0)   (1) on on ˆ= θ θ ; Output: (1) on on ˆ= D D ; (1) on on ˆ= θ θ ;

Simulation Results
In order to examine the performance of the proposed ASDPD method, we compare it with the decoupled DPD approach in [4] and covariance-based sparse DPD (CDPD) method [9].Consider four BSs placed at the corners of a 1 km × 1 km square.Assume the number of grids in the location area is 26 26 K = × , which means yielding a 40m resolution along both x and y axes.The carrier frequency of the simulated signal is assumed to be 900 MHz.Each BS is equipped with a uniform linear array of ten antenna elements with the adjacent elements spacing of half a wavelength.The locations of targets are selected at random, uniformly, within the square.All the simulation results are obtained based on 200 Monte Carlo realizations.In each simulation, we consider the following multipath channel model 1 ( ) ( ) to obtain a set of channel data. .b = 1/16 is the exponential power delay profile and i τ is the delay spread for the ith path [15].As shown in Figure 1, the improvement in location accuracy for the proposed method can be seen in terms of the root mean square error (RMSE), when the number of emitters is two and the SNR is set to 5 dB.We can observe that the location performance of DPD and CDPD algorithms decreases evidently as the number of multipath increases.On the contrary, the variation of RMSE in the ASDPD algorithm is very small due to its adaptive ability through DL technique.This result reveals that our method is very robust to multipath channels and effectively enhances location accuracy.
Figure 2 illustrates the location error with respect to the number of emitters when the SNR is set to 5 dB.Here, real lines describe the case of single-path channel for three algorithms, while dashed lines represent the case of three paths.With the increase in the number of emitters, the RMSE of DPD algorithm increases quickly due to the high sensitivity to the estimated number of targets.Note that the CDPD method does not rely on a good estimate of the number of emitters in the single-case, but its performance decreases evidently as the number of multipath increases.On the contrary, the ASDPD algorithm is very robust to two scenarios.The importance of the low sensitivity of our algorithm to the number of targets is twofold: first, the number of sources is usually unknown, and second low sensitivity provides robustness against mistakes in estimating the number of targets.

Conclusion
In this paper, we exploit the inherent spatial sparsity to present a novel direct location method by combining the offline training and online learning into a unified DL framework, thereby better matching time-varying scenarios.The effectiveness of the proposed scheme has been demonstrated by simulation results where substantial improvement for localization performance is achieved.Further research will emphasize on the off-grid error analysis and the theoretic bound on the location estimation precision.

I
, ⊗ and CN denote conjugate transpose (Hermitian), 0 l norm, 1 l norm, 2 l norm, identity matrix with the dimension N, the Kronecker product and complex Gaussian distribution, respectively.For any matrix Y, vec( ) Y is denoted as the vertical concatenation of the columns of Y. Finally, x denotes the estimate of the parameter of interest x .
set the training sample set; generate the ideal dictionary H Offline DL stage: Input: the training sample set; (0) off ˆ= D H ; the number of iterations T;for j=1 to T 1) use the reweighted 1-norm algorithm to compute the vector the gradient projection algorithm to minimize the objective function in(8) with respect to

Figure 1 .
Figure 1.The localization error with respect to the number of multipaths.

Figure 2 .
Figure 2. The localization error with respect to the number of emitters.