Efficient blind adaptive Karhunen–Loéve transform via parallel search

Abstract

This article introduces a new algorithm that constructs an efficient search strategy, called parallel search, for blind adaptive Karhunen–Loéve transform. Unlike anterior Karhunen–Loéve transform, the proposed algorithm converges quickly by searching for solutions in different directions simultaneously. Moreover, the process is “blind,” which means that minimal information about the original data is used. The new algorithm also avoids repeating the Karhunen–Loéve transform basis learning step in data compression applications. Numerical simulation results verify the validity of the theory and illustrate the capability of the proposed algorithm.

Introduction

The Karhunen–Loéve transform (KLT), also called principal component analysis (PCA),^1,2 is a well-known optimum transform for signal compression. Its basis functions are generated via a given signal covariance matrix.^3,4 Recently, KLT has been widely used in many fields, such as communication,^5,6 radar, ⁷ sparse process representations,^8–10 and distributed sensor networks. ¹¹

However, the KLT basis functions are data dependent, that is, as the signal varies, the learning process of the KLT basis function benefits from additional test data. Thus, the KLT basis functions are generated with a high computational cost while increasing the number of transmissions associated with each new basis function. As a result, the direct use of traditional KLT in data compression applications is limited.

Davila ³ proposed a scheme for blind adaptive estimation of KLT basis vectors, which can be processed by both the sender and the receiver concurrently using the same initial conditions. It can also track the KLT basis vectors by only using knowledge about the low-dimensional KLT coefficients, while not requiring explicit knowledge about the actual signal. In this sense, it is a “blind” and “adaptive” estimation scheme for KLT basis vectors.

Let ${\mathbf{x}}_n\in {\mathrm{ℝ}}^N$ be an N-dimensional column vector of the signal frame to be transmitted. Its correlation matrix $\mathbf{R}=E[{\mathbf{x}}_n{\mathbf{x}}_n^{\mathrm{T}}]$ having rank $r\le N$ has the eigenvectors denoted by $\mathbf{Q}$ whose columns are the KLT basis vectors. The traditional KLT coefficients are given by $\mathbf{y}={\mathbf{Q}}^{\mathrm{T}}{\mathbf{x}}_n$ . In practice, the eigenstructure of most signals tends to vary considerably over time. So, the KLT basis matrix needs to be constantly relearned and retransmitted within the traditional KLT framework. Davila substitutes the rank-1 updating form of the correlation matrix for $E[{\mathbf{x}}_n{\mathbf{x}}_n^{\mathrm{T}}]$ , that is

$${\hat{\mathbf{R}}}_n=\gamma {\hat{\mathbf{R}}}_{n-1}+{\mathbf{x}}_n{\mathbf{x}}_n^{\mathrm{T}}$$

(1)

where $\gamma \in (0,1)$ . ${\hat{\mathbf{Q}}}_n$ and ${\hat{\mathbf{\Lambda }}}_n$ are the estimates of the eigenvectors and eigenvalues of ${\hat{\mathbf{R}}}_n$ , respectively. ${\mathbf{I}}_N$ is the $N\times N$ identity matrix. Let $\mathbf{V}$ be an $N\times M$ stochastic matrix with independent and identically distributed (i.i.d.) Gaussian elements. The search direction vector ${\mathbf{v}}_n$ is a white noise vector independent of ${\mathbf{x}}_n$ . The iterative algorithm below shows the framework of the eigenvector estimates.

Note that if eigenvectors are merged by a search direction vector in each iteration, then the eigenvectors of ${\hat{\mathbf{R}}}_n$ can be tracked well. Davila ³ proved that the algorithm is convergent with the rate $(N-r-1)/(N-r)$ . For the sinusoidal signal used in Davila, ³ more than 300 iterations were needed to ensure that the mean squared error (MSE) between the original and recovered signals is less than ${10}^{-3}$ when $N=64$ . If $N=32$ , then the conditions are better, but 100 iterations are still needed to ensure a precise reconstruction. A well-known fact is that the smaller the selected N is, the more the signal frames are generated and the more the cost is paid due to frequent transmissions. In addition, the numerous iteration times for convergence may lead to a large amount of information loss, especially for online tracking. In fact, there is a lot of room for the convergence rate to improve. Naturally, one could ask the following question: Can we design a new scheme that retains the blind and adaptive qualities while converging faster?

In this work, we develop a new scheme that can solve this problem by introducing a parallel search.

Blind adaptive KLT via parallel search

Algorithm

The proposed scheme shown in Algorithm 2 has a similar form to the blind adaptive Karhunen–Loéve transform (BAKLT) method shown in Algorithm 1. The key difference is that the search directions are distinct. This novel scheme merges the set of estimated eigenvectors ${\hat{\mathbf{Q}}}_n$ with an N × (L – r)-dimensional matrix ${\mathbf{V}}_n$ , the column vectors of which are called parallel search vectors, and each column of ${\mathbf{V}}_n$ represents an independent search direction. Of course, if $L=r+1$ , then BAKLT is a special case of the proposed algorithm.

Algorithm 1.

Blind adaptive KLT (BAKLT).

Sender

Receiver

${\hat{\mathbf{Q}}}_0={\mathbf{I}}_N(:,1:r)$ for n = 1, 2, …  for m = 1, …, M    ${\mathbf{v}}_n=\mathbf{V}(:,m)$    ${\overline{\mathbf{Q}}}_n\stackrel{\mathrm{\Delta }}{=}[\begin{array}{cc}{\hat{\mathbf{Q}}}_{n-1}\hfill & {\mathbf{v}}_n\hfill \end{array}]$    ${\mathbf{y}}_n={\overline{\mathbf{Q}}}_{n-1}^{\mathrm{T}}{\mathbf{x}}_n$    $\mathbf{F}\stackrel{\mathrm{\Delta }}{=}\gamma {\overline{\mathbf{Q}}}_n^{\mathrm{T}}{\hat{\mathbf{Q}}}_{n-1}{\hat{\mathbf{\Lambda }}}_{n-1}{\hat{\mathbf{Q}}}_{n-1}^{\mathrm{T}}{\overline{\mathbf{Q}}}_n+{\mathbf{y}}_n{\mathbf{y}}_n^{\mathrm{T}}$    $\mathbf{G}\stackrel{\mathrm{\Delta }}{=}{\overline{\mathbf{Q}}}_n^{\mathrm{T}}{\overline{\mathbf{Q}}}_n$   solve $\mathbf{F}{\mathbf{W}}_n=\mathbf{G}{\mathbf{W}}_n{\mathbf{\Pi }}_n$    ${\hat{\mathbf{Q}}}_n={\overline{\mathbf{Q}}}_n{\mathbf{W}}_n(1:r)$   retain ${\hat{\mathbf{Q}}}_n$ , $m_0=m$ , and ${\mathbf{y}}_n$ if $\mathrm{tr}{\mathbf{\Pi }}_n(1:r,1:r)$ is maximum  end  transmit ${\mathbf{y}}_n$ , $m_0$ end

${\hat{\mathbf{Q}}}_0={\mathbf{I}}_N(:,1:r)$ for n = 1, 2, …  wait for ${\mathbf{y}}_n$ , $m_0$    ${\mathbf{v}}_n=\mathbf{V}(:,m_0)$    ${\overline{\mathbf{Q}}}_n=[\begin{array}{cc}{\hat{\mathbf{Q}}}_{n-1}\hfill & {\mathbf{v}}_n\hfill \end{array}]$       $\mathbf{F}=\gamma {\overline{\mathbf{Q}}}_n^{\mathrm{T}}{\hat{\mathbf{Q}}}_{n-1}{\hat{\mathbf{\Lambda }}}_{n-1}{\hat{\mathbf{Q}}}_{n-1}^{\mathrm{T}}{\overline{\mathbf{Q}}}_n+{\mathbf{y}}_n{\mathbf{y}}_n^{\mathrm{T}}$    $\mathbf{G}={\overline{\mathbf{Q}}}_n^{\mathrm{T}}{\overline{\mathbf{Q}}}_n$   solve $\mathbf{F}{\mathbf{W}}_n=\mathbf{G}{\mathbf{W}}_n{\mathbf{\Pi }}_n$    ${\hat{\mathbf{Q}}}_n={\overline{\mathbf{Q}}}_n{\mathbf{W}}_n(1:r)$  output ${\hat{\mathbf{x}}}_n={\hat{\mathbf{Q}}}_{n-1}{\mathbf{y}}_n(1:N)$ end

Algorithm 2.

Blind adaptive KLT via parallel search.

Sender

Receiver

${\hat{\mathbf{Q}}}_0={\mathbf{I}}_N(:,1:r)$ for n = 1, 2, …  for m = L – r, …, M    ${\mathbf{V}}_n=\mathbf{V}(:,m-L+r+1:m)$    ${\overline{\mathbf{Q}}}_n\stackrel{\mathrm{\Delta }}{=}[\begin{array}{cc}{\hat{\mathbf{Q}}}_{n-1}\hfill & {\mathbf{V}}_n\hfill \end{array}]$    ${\mathbf{y}}_n={\overline{\mathbf{Q}}}_{n-1}^{\mathrm{T}}{\mathbf{x}}_n$    $\mathbf{F}\stackrel{\mathrm{\Delta }}{=}\gamma {\overline{\mathbf{Q}}}_n^{\mathrm{T}}{\hat{\mathbf{Q}}}_{n-1}{\hat{\mathbf{\Lambda }}}_{n-1}{\hat{\mathbf{Q}}}_{n-1}^{\mathrm{T}}{\overline{\mathbf{Q}}}_n+{\mathbf{y}}_n{\mathbf{y}}_n^{\mathrm{T}}$    $\mathbf{G}\stackrel{\mathrm{\Delta }}{=}{\overline{\mathbf{Q}}}_n^{\mathrm{T}}{\overline{\mathbf{Q}}}_n$   solve $\mathbf{F}{\mathbf{W}}_n=\mathbf{G}{\mathbf{W}}_n{\mathbf{\Pi }}_n$    ${\hat{\mathbf{Q}}}_n={\overline{\mathbf{Q}}}_n{\mathbf{W}}_n(:,1:r)$   retain ${\hat{\mathbf{Q}}}_n$ , $m_0=m$ , and ${\mathbf{y}}_n$ if $\mathrm{tr}{\mathbf{\Pi }}_n(1:r,1:r)$ is maximum  end  transmit ${\mathbf{y}}_n$ , $m_0$ , L end

${\hat{\mathbf{Q}}}_0={\mathbf{I}}_N(:,1:r)$ for n = 1, 2, …  wait for ${\mathbf{y}}_n$ , $m_0$ , L    ${\mathbf{V}}_n=\mathbf{V}(:,m_0-L+r+1:m_0)$    ${\overline{\mathbf{Q}}}_n=[\begin{array}{cc}{\hat{\mathbf{Q}}}_{n-1}\hfill & {\mathbf{V}}_n\hfill \end{array}]$    $\mathbf{F}=\gamma {\overline{\mathbf{Q}}}_n^{\mathrm{T}}{\hat{\mathbf{Q}}}_{n-1}{\hat{\mathbf{\Lambda }}}_{n-1}{\hat{\mathbf{Q}}}_{n-1}^{\mathrm{T}}{\overline{\mathbf{Q}}}_n+{\mathbf{y}}_n{\mathbf{y}}_n^{\mathrm{T}}$    $\mathbf{G}={\overline{\mathbf{Q}}}_n^{\mathrm{T}}{\overline{\mathbf{Q}}}_n$   solve $\mathbf{F}{\mathbf{W}}_n=\mathbf{G}{\mathbf{W}}_n{\mathbf{\Pi }}_n$    ${\hat{\mathbf{Q}}}_n={\overline{\mathbf{Q}}}_n{\mathbf{W}}_n(:,1:r)$  output ${\hat{\mathbf{x}}}_n={\hat{\mathbf{Q}}}_{n-1}{\mathbf{y}}_n(1:r)$ end

The colon (by itself) refers to all the elements in a row or a column of a matrix. Thus, ${\mathbf{I}}_N(:,1:r)$ means the column 1 through r of the matrix ${\mathbf{I}}_N$ .

We demonstrate that the proposed algorithm is convergent and faster, while the proof is simpler.

Convergence

To measure the algorithm’s performance, the a priori MSE between the original and reconstructed data frames is estimated as

$$\epsilon \left(n\right)=\Vert \mathbf{x}-{\hat{\mathbf{Q}}}_n{\hat{\mathbf{Q}}}_n^{\mathrm{T}}\mathbf{x}{\Vert }^2$$

(2)

where ‖·‖ is the standard vector 2-norm. Since ${\hat{\mathbf{Q}}}_n$ contains orthonormal vectors, then ${\hat{\mathbf{Q}}}_n{\hat{\mathbf{Q}}}_n^{\mathrm{T}}$ is the projection matrix ${\mathbf{P}}_{{\hat{\mathbf{Q}}}_n}$ onto $\Re ({\hat{\mathbf{Q}}}_n)$ . The expected value of the error in equation (2) can be written as

$$E\left[\epsilon \left(n\right)\right]=E\left[\mathrm{tr}\left({\mathbf{x}}^{\mathrm{T}}{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_n}^{\perp }\mathbf{x}\right)\right]$$

(3)

where ${\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_n}^{\perp }={\mathbf{I}}_N-{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_n}$ is the orthogonal projection matrix. We can write

$${\mathbf{P}}_{{\overline{\mathbf{\Phi }}}_n}^{\perp }=I_N-{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{n-1}}^{\perp }-{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{n-1}}^{\perp }{\mathbf{V}}_n{\left({\mathbf{V}}_n^{\mathrm{T}}{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{n-1}}^{\perp }{\mathbf{V}}_n\right)}^{-1}{\mathbf{V}}_n^{\mathrm{T}}{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{n-1}}^{\perp }$$

(4)

given that ${\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_n}^{\perp }={\mathbf{P}}_{{\overline{\mathbf{\Phi }}}_n}^{\perp }$ . ³ Therefore, we can say

$$\begin{array}{c}\multicolumn{1}{c}{E\left[\epsilon \left(k\right)\right]=\mathrm{tr}{\mathbf{x}}^{\mathrm{T}}·E\{{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{n-1}}^{\perp }-\left[{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{n-1}}^{\perp }{\mathbf{V}}_n{\left({\mathbf{V}}_n^{\mathrm{T}}{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{n-1}}^{\perp }{\mathbf{V}}_n\right)}^{-1}{\mathbf{V}}_n^{\mathrm{T}}{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{n-1}}^{\perp }\right]\}\mathbf{x}}\\ =\mathrm{tr}{\mathbf{x}}^{\mathrm{T}}{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{n-1}}^{\perp }\{\mathbf{I}-E\left[{\mathbf{V}}_n{\left({\mathbf{V}}_n^{\mathrm{T}}{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{n-1}}^{\perp }{\mathbf{V}}_n\right)}^{-1}{\mathbf{V}}_n^{\mathrm{T}}{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{n-1}}^{\perp }\right]\}\mathbf{x}\hfill \\ =\mathrm{tr}{\mathbf{x}}^{\mathrm{T}}{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{n-1}}^{\perp }\mathbf{x}\times \{1-E\left[\frac{{\mathbf{x}}^{\mathrm{T}}{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{n-1}}^{\perp }{\mathbf{V}}_n{\left({\mathbf{V}}_n^{\mathrm{T}}{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{n-1}}^{\perp }{\mathbf{V}}_n\right)}^{-1}{\mathbf{V}}_n^{\mathrm{T}}{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{n-1}}^{\perp }\mathbf{x}}{{\mathbf{x}}^{\mathrm{T}}{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{n-1}}^{\perp }\mathbf{x}}\right]\}\hfill \end{array}$$

(5)

Let ${\mathbf{X}}_k={\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{k-1}}^{\perp }{\mathbf{V}}_k$ ; then each element of ${\mathbf{X}}_k$ follows a Gaussian distribution because ${\mathbf{V}}_k$ has a standard normal distribution, while ${\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{k-1}}^{\perp }$ is a linear transform. We note that

$$E\left[{\mathbf{X}}_k{\left({\mathbf{X}}_k^{\mathrm{T}}{\mathbf{X}}_k\right)}^{-1}{\mathbf{X}}_k^{\mathrm{T}}\right]=\sigma {\mathbf{I}}_N$$

(6)

is formally satisfied, where $\sigma$ is unknown. We can also write

$$\begin{array}{c}\mathrm{tr}\left\{E\left[{\mathbf{X}}_k{\left({\mathbf{X}}_k^{\mathrm{T}}{\mathbf{X}}_k\right)}^{-1}{\mathbf{X}}_k^{\mathrm{T}}\right]\right\}=\sigma ·\mathrm{tr}\left({\mathbf{I}}_N\right)\hfill \\ =E\left\{\mathrm{tr}\left[{\left({\mathbf{X}}_k^{\mathrm{T}}{\mathbf{X}}_k\right)}^{-1}{\mathbf{X}}_k^{\mathrm{T}}{\mathbf{X}}_k\right]\right\}=L-r\hfill \\ \iff \sigma ·N=L-r\hfill \end{array}$$

(7)

Using equations (3), (5), and (7), the expectation error can be written as

$$\begin{array}{c}\multicolumn{1}{c}{\epsilon \left(k\right)=\epsilon (k-1)\times \{1-\sigma \left[\frac{{\mathbf{x}}^{\mathrm{T}}{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{k-1}}^{\perp }\mathbf{x}}{{\mathbf{x}}^{\mathrm{T}}{\mathbf{P}}_{{\hat{\mathbf{\Phi }}}_{k-1}}^{\perp }\mathbf{x}}\right]\}}\\ =\epsilon (k-1)(1-\frac{L-r}{N})\hfill \end{array}$$

(8)

We note that the convergence rate correlates with the ratio between the number of parallel search directions and the signal length. Obviously, when $L=r+1$ , the error reduction ratio of BAKLT in Davila ³ is $1/N$ . Thus, the bigger the frame length is, the slower the algorithm convergence is. The proposed algorithm increases the convergence rate to $(L-r)/N$ via parallel search, which is $(L-r)$ times faster than BAKLT.

Numerical experiments and results

In this section, we experimentally illustrate that the parallel search is a powerful approach for accelerating the convergence, when compared to BAKLT. In our simulations, the signal of interest is generated as in Davila, ³ that is

$$\begin{array}{c}x\left(n\right)=\hfill \\ \{\begin{array}{c}\cos (0.35\pi n)+\cos (0.78\pi n+0.35\pi ),n=1,\dots ,999\hfill \\ \multicolumn{1}{c}{\cos (0.6\pi n)+\cos (0.8\pi n+0.35\pi ),n=1000,\dots ,2000}\end{array}\hfill \end{array}$$

(9)

when the algorithm is applied with $r=4$ and $\gamma =0.8$ . The parallel search vectors ${\mathbf{V}}_n$ are selected from an $N\times 1000$ zero-mean Gaussian white noise codebook $\mathbf{V}$ , which is shared by the sender and the receiver, to find the best search directions.

Estimation error

The major goal of the first experiment is to validate the performance of the algorithm (to recover a signal). In this experiment, the MSE $\epsilon (n)$ is defined as in equation (2). To verify that the KLT basis vectors were indeed being tracked, the eigenvectors of the sample correlation matrix were computed and substituted for the matrix ${\mathbf{Q}}_n^o$ . The measure of the distance between the subspace spanned by the estimated KLT basis vectors in ${\hat{\mathbf{Q}}}_n$ and the eigenvectors in ${\mathbf{Q}}_n^o$ is given by

$$\xi \left(n\right)=\Vert {\mathbf{P}}_{{\mathbf{Q}}_n^o}-{\mathbf{P}}_{{\hat{\mathbf{Q}}}_n}{\Vert }_F$$

(10)

where $\Vert ·{\Vert }_F$ denotes the matrix Frobenius norm. ${\mathbf{P}}_{{\mathbf{Q}}_n^o}$ and ${\mathbf{P}}_{{\hat{\mathbf{Q}}}_n}$ are the projection matrices onto the column spaces of ${\mathbf{Q}}_n^o$ and ${\hat{\mathbf{Q}}}_n$ , respectively.

To ensure the accuracy of the calculation, the Multiprecision Computing Toolbox for MATLAB ¹² is used.

Figure 1 illustrates the signal reconstruction MSE results $\epsilon (n)$ for different measurement lengths L using the parallel search method when N = 16. We can see that the proposed algorithm is convergent based on the number of iterations it completes. Under the same conditions, the more parallel the search directions are (i.e. the bigger the length L measurements are), the faster the MSE converges to zero. Thus, the parallel search method leads to a significant increase in convergence rate. Figure 2 shows the $\xi (n)$ function. This demonstrates that the algorithm can indeed track the KLT basis vectors. Therefore, the changing trends of $\xi (n)$ are similar to those of $\epsilon (n)$ , as shown in Figure 1. Compared with BAKLT (i.e. $L=r+1$ ), the parallel search leads to different significant increases in convergence rate for different measurement lengths.

Figure 1.

Mean squared error for signal reconstruction at different measurement lengths L using the parallel search method when $N=16$ . The parallel search method leads to a significant improvement of the convergence rate.

Figure 2.

KLT tracking error for different measurement lengths L using the parallel search when $N=16$ .

To analyze the relationship between the convergence rate and measurement lengths L, another experiment is designed.

Convergence rate

In this experiment, the parallel search algorithm is convergent when the MSE $\epsilon (n)$ , between the reconstruction and the original signal, is smaller than 10⁻³ or when $\Vert \mathbf{x}-{\hat{\mathbf{Q}}}_n{\hat{\mathbf{Q}}}_n^{\mathrm{T}}\mathbf{x}{\Vert }^2<{10}^{-3}$ . The number of iterations needed for the parallel search to converge is compared to the signal frame length L when $N=16,32,\mathrm{and}64$ , respectively. The results are shown in Figure 3. Thus, the parallel search provides a considerable reduction in the number of iterations needed for convergence. When $N=64$ and $L=r+1$ corresponded to BAKLT, 341 iterations are needed for the process to converge because it searches in only one direction at a time. If $N=64$ and $L=32$ , only 54 iterations are required, which benefits from the $L-r$ parallel direction search in each iteration. In addition, the frame length N is inversely proportional to the recovered MSE. The convergence rate also varies inversely with N, which can be seen in the convergence analysis in section “Blind adaptive KLT via parallel search.” We note that the frame length N determines the frequency of transmission if the signal length is fixed, so N should not be too small a value. The parallel search method is useful when accelerating the convergence rate and improving the reconstruction accuracy. Our analysis shows that there is a trade-off between measurement length and convergence rate. The more parallel the measurements are, the faster the convergence rate is. In practice, a balanced selection process is needed, but the compression and convergence speed must also be taken into consideration. The blind adaptive KLT via our parallel search method may provide this additional choice.

Figure 3.

The number of iterations for convergence versus different measurement lengths L using the parallel search process when $N=64,32,\mathrm{and}16$ , respectively.

Conclusion

In this article, we proposed a novel algorithm that constructs a fast and efficient search strategy for blind adaptive KLT. The algorithm, called parallel search, convergences quickly due to its reduced processing steps. We have proven that, for Gaussian white noise parallel search vectors, the convergence is dependent on the quantity $(L-r)/N$ , where N is the data frame length and r is the number of KLT basis vectors. Thus, the larger the $(L-r)/N$ term, the faster the convergence. In addition, the parallel search increases the convergence rate $(L-r)$ times compared to that of BAKLT. In practice, a workable balance can be found between the compression and convergence speed by adjusting the measurement lengths.