 Research
 Open access
 Published:
Semitensor productbased onebit compressed sensing
EURASIP Journal on Advances in Signal Processing volumeÂ 2023, ArticleÂ number:Â 112 (2023)
Abstract
The area of onebit compressed sensing (1bit CS) focuses on the recovery of sparse signals from binary measurements. Over the past decade, this field has witnessed the emergence of welldeveloped theories. However, most of the existing literature is confined to fully random measurement matrices, like random Gaussian and random subGaussian measurements. This limitation often results in high generation and storage costs. This paper aims to apply semitensor productbased measurements to 1bit CS. By utilizing the semitensor product, this proposed method can compress highdimensional signals using lowerdimensional measurement matrices, thereby reducing the cost of generating and storing fully random measurement matrices. We propose a regularized model for this problem that has a closedform solution. Theoretically, we demonstrate that the solution provides an approximate estimate of the underlying signal with upper bounds on recovery error. Empirically, we conduct a series of experiments on both synthetic and realworld data to demonstrate the proposed methodâ€™s ability to utilize a lowerdimensional measurement matrix for signal compression and reconstruction with enhanced flexibility, resulting in improved recovery accuracy.
1 Introduction
Compressed (CS) has drawn widely studied and applied in many fields such as moderate resolution imaging [1, 2] and magnetic resonance imaging (MRI). Onebit compressed sensing (1bit CS) [3,4,5,6,7,8,9] has garnered significant attention in recent years as a unique paradigm of (CS) [10, 11]. In contrast to the classic compressed sensing method, which utilizes an infiniteprecision realvalued measurement \({{\varvec{b}}}={{\varvec{A}}}{{{{\varvec{x}}}}}\) where \({\varvec{A}}\in {\mathbb{R}}^{m\times n}\) is the measurement matrix and \({{\varvec{x}}}\in {\mathbb{R}}^{n}\) denotes a sparse signal, 1bit CS only records the signs of realvalued measurements as \({{\varvec{y}}}={\text{sign}}({\varvec{b}})={\text{sign}}({\varvec{A}}{{\varvec{x}}})\). Using binary measurement, 1bit CS offers a costeffective hardware implementation and highspeed sampling. Additionally, it exhibits greater resilience to various nonlinear distortions commonly encountered in input electronics [5]. Due to the aforementioned reasons, it possesses a wide range of potential applications in massive MIMO systems [12, 13], wireless sensor networks [14], synthetic aperture radar systems [15], and other scenarios where largescale sparse data are typically involved [16].
Mathematically, standard 1bit CS aims to recover the direction of the underlying sparse signal \({{\varvec{x}}}\in {\mathbb{R}}^{n}\) (i.e., \({{\varvec{x}}}/\Vert {{\varvec{x}}}\Vert _{2}\)) from 1bit measurements
where \({\text{sign}}(\cdot )\) applies componentwise on realvalued measurement \({\varvec{A}}{{\varvec{x}}}\), satisfying \({\text{sign}}(v)=1\) if \(v\ge 0\), and \({\text{sign}}(v)=1\) if \(v<0\). The standard measurement model (1) fails to capture information on the magnitude of \({{\varvec{x}}}\) (i.e., \(\Vert {{\varvec{x}}}\Vert _{2}\)), resulting in the aforementioned studies being limited to the unit sphere \({\mathbb{S}}_{n}:=\{{{\varvec{x}}}\in {\mathbb{R}}^{n}:\Vert {{\varvec{x}}}\Vert _{2}=1\}\). Jacques et al. [5] have proposed an \(\ell _{0}\)minimization method for this problem, which has been proven to yield an optimal solution that provides a robust estimate of the original signal. Although a binary iterative hard threshold (BIHT) algorithm has been developed to approximate the underlying solution empirically, solving the \(\ell _{0}\)minimization model exactly remains an NPhard problem. Plan and Vershynin [6] utilize the \(\ell _{1}\)norm as a means of promoting sparsity, proposing a convex programming approach that is also supported by theoretical guarantees. Moreover, their method demonstrates robustness against bit flips, a prevalent form of noise in binary measurements. Zhang et al. [7] suggest the following regularized model
and present the existence of a closedform solution, namely \({\mathcal {S}}_{\gamma }({\varvec{A}}^{T}{\varvec{y}})/\Vert {\mathcal {S}}_{\gamma }({\varvec{A}}^{T}{\varvec{y}})\Vert _{2}\) where \({\mathcal {S}}_{\gamma }(\cdot )\) denotes the softthresholding operator [psee Eq. (22)]. However, the \(\ell _{1}\)norm is a loose approximation of \(\ell _{0}\)quasinorm, often resulting in an overpenalized problem [17]. In our prior research [8], we proposed an \(\ell _{p}\)(\(0<p<1\))minimization method and demonstrated its superiority over the \(\ell _{1}\)norm based counterpart in terms of reducing the required number of measurements.
The aforementioned studies have solely focused on fully random Gaussian distributions, which limits their applicability in processing largescale data. Unfortunately, Ai et al. [18] have shown that even with the commonly used fully random subGaussian measurement matrices, it is impossible to recover any sparse unit signal from (1) without additional constraints (e.g., \({{\varvec{x}}}\) is not too sparse). To mitigate this issue, it has recently been discovered that introducing dithers can be beneficial. Let
where \({{\varvec{\tau }}}\in {\mathbb{R}}^{m}\) denotes a random dither vector. Dirksen and Mendelson [19] demonstrated that a convex program can be applied to recover the signal \({{\varvec{x}}}\in {\mathbb{R}}^{n}\) from (3) when \({\varvec{\tau }}\) is generated from a uniform distribution, and \({\varvec{A}}\) is a random subGaussian measurement matrix. However, fully subGaussian measurements still pose challenges in terms of high costs for generating and storing measurement matrices. Dirksen et al. [20] showed that randomly subsampled Gaussian circulant matrices can be utilized for 1bit CS, and Liu et al. [21] provide an optimization algorithm based on it under generative prior. Dirksen et al. [22] further applied randomly subsampled subGaussian circulant matrices to 1bit CS and prove that the proposed algorithm is robust to analog prequantization noise as well as to adversarial bit corruptions in the quantization process. It should be noted that circulant measurement matrices are typically limited to subsampling, while oversampling is commonly employed in 1bit CS despite being irrelevant in traditional CS problems. In practical applications of circulant matrices, we have to introduce the zeropadding technique on the original signal to accommodate oversampling, which lifts the original signal to a higher dimension. Additionally, a square measurement matrix with a matching dimension has to be constructed. In summary, the research on designing measurement matrices for 1bit CS is still in its early stages.
The concept of semitensor product of matrices was proposed by Cheng et al., which is a generalization of conventional matrix product [23,24,25]. The introduction of semitensor product enables matrix multiplication even when two matrices do not meet the dimension matching condition. This concept has found applications in various engineering fields, including but not limited to information security [26], vehicle control [27], and gene regulation [28]. Specially, Xie et al. [29] proposed a new model of signal compression and reconstruction based on the semitensor product, which breaks the dimension matching conditions required by the traditional CS model. The novel approach employs lowerdimensional sensing matrices to compress highdimensional signals, while simultaneously reducing total reconstruction time by conducting the reconstruction phase among multiple CS decoders. Inspired by these advantages, we are considering the potential application of semitensor product in 1bit CS to tackle the challenge of storing large measurement matrices. The primary contributions of this paper are as follows.

We first apply the semitensor product to the measurement model of 1bit CS with employing the dither technique. The novel measurement model significantly reduces the dimensionality of the measurement matrix compared to existing fully random counterparts in 1bit CS.

We propose a regularized model with a closedform solution and demonstrate that our proposed model, which employs an explicit selection strategy for the regularization parameter, can yield an approximate estimate of the underlying sparse signal with guaranteed convergence.
The rest of this paper is structured as follows. SectionÂ 2 introduces some notations, definitions, and useful lemmas. SectionÂ 3 clarifies the measurement model used in this paper. SectionÂ 4 elucidates the recovery model and then gives the corresponding theoretical guarantee. SectionÂ 5 presents numerical experiments, and Sect.Â 6 concludes this paper.
2 Preliminaries
2.1 Notations
We begin with some notations used throughout the paper. Scalars are denoted with lowercase letters, e.g., x, vectors with boldface lowercase letters, e.g., \({{\varvec{x}}}\) and matrices with boldface capital letters, e.g., \(\varvec{X}\). We denote \(x_i\) as the ith element of the vector \({{\varvec{x}}}\). The fields of real numbers are denoted as \({\mathbb{R}}\). [n] denotes the index set \(\{1,2,\ldots ,n\}\). \(\varvec{I}_{n}\) denotes identity matrix of size \(n\times n\). The inner product between matrices \(\varvec{M}=(m_{ij})\in {\mathbb{R}}^{n_{1}\times n_{2}}\) and \(\varvec{N}=(n_{ij})\in {\mathbb{R}}^{n_{1}\times n_{2}}\) of the same size is defined as \(\langle \varvec{M},\varvec{N}\rangle ={\text{Tr}}({\varvec{M}}^{T}\varvec{N})=\sum _{ij}m_{ij}n_{ij}\). We define the \(\ell _{p}\) norm of \({{\varvec{x}}}\in {\mathbb{R}}^{n}\) as \(\Vert {{\varvec{x}}}\Vert _{p}=(\sum _{i=1}^{n}x_{i}^{p})^{1/p}\) where \(p\ge 1\). \(\Vert {{\varvec{x}}}\Vert _{\infty }\) denotes the absolute value of the elements of \({{\varvec{x}}}\) that has the biggest magnitude.
2.2 Semitensor product and some definitions
Definition 1
(See [25]) Let \(\varvec{u}\in {\mathbb{R}}^{1\times nl}\) be a row vector, and \(\varvec{v}\in {\mathbb{R}}^{l}\) be a column vector. Divide \(\varvec{u}\) into l contiguous blocks, named \(\varvec{u}_{1},\ldots ,\varvec{u}_{l}\), which are all row vectors of length n. Then, the semitensor product (STP) between \(\varvec{u}\) and \(\varvec{v}\), denoted by \(\varvec{u}\ltimes \varvec{v}\), is defined as
Definition 2
(See [25]) Let \({\varvec{A}}\in {\mathbb{R}}^{m\times n}\), \(\varvec{B}\in {\mathbb{R}}^{p\times q}\), \(\varvec{a}_{i}\) denote the ith row of \({\varvec{A}}\), and \({\varvec{b}}^{i}\) denote the ith column of \(\varvec{B}\). The STP of \({\varvec{A}}\) and \(\varvec{B}\), denoted as \(\varvec{C}={\varvec{A}}\ltimes \varvec{B}\), consists of \(m\times q\) blocks as \(\varvec{C}=(\varvec{c}_{ij})\) where

(i)
\(\varvec{c}_{ij}=\varvec{a}_{i}\ltimes {\varvec{b}}^{i}\) is of length p when p is a factor of n, and

(ii)
\(\varvec{c}_{ij}=({{\varvec{b}}^{i}}^{T}\ltimes \varvec{a}_{i}^{T})^{T}\) is of length n when n is a factor of p,
for \(i\in [m]\) and \(j\in [q]\). Specially, when \(q=1\), \(\varvec{C}\) reduces to a vector of length mk where \(k:=gcd(n,p)\) denotes the greatest common divisor(gcd).
We also introduce the Kronecker product, which is a mathematical operation that combines two matrices to form a larger matrix with block entries.
Definition 3
The Kronecker product between matrices \({\varvec{A}}=(a_{ij})\in {\mathbb{R}}^{m\times n}\) and \(\varvec{B}\in {\mathbb{R}}^{p\times q}\) is defined as
It is readily checked that the Kronecker product has the following property.
Proposition 1
For any matrices \({\varvec{A}}\in {\mathbb{R}}^{m\times n}\) and \(\varvec{B}\in {\mathbb{R}}^{p\times q}\), it holds that
The STP of matrices can be equivalently defined through the Kronecker product.
Definition 4
(See [24]) The STP of two matrices \({\varvec{A}}\in {\mathbb{R}}^{m\times n}\) and \(\varvec{B}\in {\mathbb{R}}^{p\times q}\) satisfies
where t is the least common multiple of n and p, denoted as \(t=lcm(n,p)\).
The following lemma was proved in [7], which may facilitate our analysis.
Lemma 1
(See [7]) Suppose \({\varvec{b}}=\theta ({\varvec{A}}\varvec{u})\) with \(\theta (\cdot )\) satisfying (10), and \({\varvec{A}}\in {\mathbb{R}}^{m\times n}\) is populated by i.i.d. random standard Gaussian variables. With probability at least \(1e^{1t}\), it holds that
3 Problem description
Let \({\varvec{A}}\) be the measurement matrix of size \(m\times n\). We suppose the signal \({\varvec{x}}\) is of length nk where \(k\ge 1\) denotes some integer. As a result, the measurement matrix only requires 1/k of the parameters compared to traditional sensing models, which typically have a size of \(m\times nk\). The standard measurement model for 1bit CS under STP may be naturally given by
where \(y\in \{1,1\}^{mk}\) denotes the sign measurements. Note that (5) loses any scaling information, and is restricted to narrow types of measurement such as random Gaussian distribution. To circumvent this issue, a mature strategy is to introduce random dither \({\varvec{\tau }}\in {\mathbb{R}}^{mk}\) before quantization, leading to the dithered onebit measurements
where \(y\in \{1,1\}^{mk}\). Please see Sect.Â 4.3 and [3] for more discussion on the dither technology. To facilitate discussion, we define the exactly sparse set
and the approximately sparse set
In practice, there will always be adversarial noise in the measurements during acquisition and transmission, which usually flip some signs [5]. In this case, the noisy measurements may be modeled as
where \(\odot\) denotes the dot product between vectors, \(\varvec{\xi }\in \{1,1\}^{mk}\) denotes random bit flips, which are populated by i.i.d. Bernoulli random variables satisfying \({{P}}(\xi _{i}=1)=p\), meaning that each sign measurement \(y_{i}\) is only correct with probability p.
To illustrate how to construct a loss function for the noisy measurements (9), inspired by [6], we begin with a more general measurement model. We assume
where \(z_{i}=(\varvec{A}\ltimes {\varvec{x}})_{i}\), \({\varvec{x}}\in {\mathbb{R}}^{n}\) denotes the neededtoberecovered signal, and \(\theta (\cdot )\) is some function unknown or unspecified, which automatically must satisfy \(1\le \theta (z)\le 1\). We also assume
to capture the relation between \(y_{i}\) and \((\varvec{A}\ltimes {\varvec{x}}+{\varvec{\tau }})_{i}\), where g is a standard normal random variable. In (9), since \(y_{i}=\xi _{i}{\text{sign}}((\varvec{A}\ltimes {\varvec{x}})_{i}+\tau _{i})\) for each \(i\in [m]\), then it holds that \(\theta (z)={E}_{\tau _{i}}({E}_{\xi _{i}}(y_{i}z,\tau _{i}))=2(p1/2)(12{P}(\tau \le z))\) in (10), and the correlation coefficient \(\lambda ={E}(\theta (g)g)\) can be evaluated using integration by parts, which gives
For instance, if \(\tau _{i}(\forall i\in [mk])\) are random variables generated from the uniform distribution
where \(c>0\) denotes a constant, then
where \(\Phi (\cdot )\) denotes the distribution function of standard Gaussian variables.
We formulate theories based on model (10) in the following section as it naturally encompasses the model presented in model (9).
4 Main results
This part introduces the optimization model for recovering the underlying signal from model (10) and subsequently establishes its corresponding theoretical guarantee.
4.1 The proposed model
Suppose \({\varvec{x}}^{*}\in {\mathbb{R}}^{nk}\) is the underlying signal in \({\mathbb{K}}_{s}\) where k is a positive integer, and \(\varvec{y}\) satisfies (10). We reorder its entries to define \(\tilde{{\varvec{x}}}_{*}\) such that
and also reorder entries of \(\varvec{y}\) to define \(\tilde{\varvec{y}}\) such that
A direction application of Lemma 4 in [7] to each \(\tilde{\varvec{y}}^{(l)}\) indicates the following result.
Lemma 2
Suppose \(\varvec{A}\in {\mathbb{R}}^{m\times n}\) is populated by i.i.d. random standard Gaussian variables and \(\varvec{a}_{i}\) denotes the ith row of \(\varvec{A}\). Following the definitions given by (10) and (11), it holds that
where \(\tilde{{\varvec{x}}}_{*}^{(l)}\), \(\tilde{\varvec{y}}^{(l)}\) are, respectively, defined as in (14), (15), and \({\tilde{y}}^{(l)}_{i}\) represents the ith element of \(\tilde{\varvec{y}}^{(l)}\).
By taking the inner product in both sides of (16) with respect to \({\tilde{{\varvec{x}}}_{*}^{(l)}}\) and then summing over all \(l\in [m]\), we can derive that
When \(\theta (\cdot )={\text{sign}}(\cdot )\), \(\lambda ={ E}(g)=\sqrt{2/\pi }\) achieves the maximum relation [6]. According to the law of large numbers, Eq.Â (17) suggests that the optimal solution \(\hat{{\varvec{x}}}\) is expected to also achieve a large value for the relation function
under the constraint \(\Vert \varvec{\hat{{\varvec{x}}}}\Vert _{2}\le 1\). It is not hard to check that
On the other hand, given the sparsity of \({\varvec{x}}\), it is imperative that \(\hat{{\varvec{x}}}\) also exhibits sparsity, which can be achieved through regularization using the \(\ell _{1}\)norm \(\Vert \cdot \Vert _{1}\). By combining (18) with the \(\ell _{1}\)norm regularizer via a regularization parameter \(\gamma >0\), we propose the optimization model
Since \(\varvec{A}\ltimes \varvec{u}=(\varvec{A}\otimes \varvec{I}_{k})\varvec{u}\) by Definition 4 and Proposition 1, we can equivalently rewrite (19) as
4.2 Theoretical guarantee
The following result shows that the model (20) has a closedform solution.
Lemma 3
(See [7]) The optimal solution to (20) is given by
where the softthresholding operator \(S_{\gamma }(\cdot )\)
applies componentwise.
To prove the main theorem, we first provide the following lemma.
Lemma 4
Suppose \(\varvec{A}\in {\mathbb{R}}^{m\times n}\) is populated by independent and identically distributed random standard Gaussian variables, \({\varvec{x}}^{*}\in {\mathbb{R}}^{nk}\) where k is a positive integer, and \(\varvec{y}=\theta (\varvec{A}\ltimes {\varvec{x}}_{*})\) with \(\theta (\cdot )\) satisfying (10). Then, with probability at least \(1ke^{1t}\),
where \(\kappa >0\) is a constant, and \(\lambda\) is specified by (11).
Proof
We define \(\tilde{{\varvec{x}}}_{*}\) and \(\tilde{\varvec{y}}_{*}\) as given in (14) and (15), respectively. For each \(l\in [k]\), we have \(\tilde{\varvec{y}}^{(l)}=\varvec{A}\tilde{{\varvec{x}}}_{*}^{(l)}\) and
Since Lemma 1 indicates that
holds with a probability at least \(1e^{1t}\) for each \(l\in [k]\), taking a union bound over all l implies that
holds with a probability at least \(1ke^{1t}\). Therefore, we complete our proof. \(\square\)
Now, we can present the main theoretical result.
Theorem 1
Assume \(\varvec{A}\in {\mathbb{R}}^{m\times n}\) is populated by independent and identically distributed random standard Gaussian variables, \(\varvec{y}\) is observed via (9), \(\lambda\) is defined in Eq.Â (13), and
where \(c=2\kappa\) is a constant with \(\kappa\) coming from (23). If \({\varvec{x}}^{*}\in {\mathbb{R}}^{nk}\) belongs to (7) where k is a positive integer, with a probability at least \(1ke^{1t}\), the estimation \(\hat{{\varvec{x}}}\) given by (21) satisfies
If \({\varvec{x}}^{*}\in {\mathbb{R}}^{nk}\) belongs to (8), with probability exceeding \(1ke^{1t}\) the estimation \(\hat{{\varvec{x}}}\) given by (21) satisfies
Proof
Since \(\hat{{\varvec{x}}}\) is the optimal solution, we have
Thus one has
Combining it with Lemma 4, we have that
The remain discussion considers the two distinct types of sparse signals, respectively.
(1) When \({\varvec{x}}_{*}\) is exactly sparse, i.e., \({\varvec{x}}_{*}\in {\mathbb{K}}_{s}^{*}\), we define S as the support set of \({\varvec{x}}_{*}\) and \(S_{\perp }=[nk]{\setminus } S\) be its complement set. We also define \(P_{S}(\varvec{u})\) as
denoting the subvector of \(\varvec{u}\) indexed by the set S. Then (26) implies
Therefore, using triangle inequality yields
where the final inequality follows the Cauchyâ€“Schwarz inequality. Recalling \(\Vert {\varvec{x}}_{*}\hat{{{\varvec{x}}}}\Vert _{2}^{2}\le 2(1\hat{{{\varvec{x}}}}^{T}{\varvec{x}}_{*})\), we obtain that
i.e., \(\Vert {\varvec{x}}_{*}\hat{{{\varvec{x}}}}\Vert _{2}\le \frac{3\gamma }{\lambda }\sqrt{s}.\)
(2) When \({\varvec{x}}_{*}\) is approximately sparse, i.e., \({\varvec{x}}_{*}\in {\mathbb{K}}_{s}\), from (26), we have
which indicates
We complete our proof. \(\square\)
Remark 1
Theorem 1 offers a practical recommendation for selecting the regularization parameter \(\gamma\) in model (20). As demonstrated experimentally in Sect.Â 5.1, when combined with finely tuned c in (25), the regularization parameter in (20) performs well on general sparse signals.
Remark 2
According to the suggestion put forward by [7], we have selected \(u=1\) in (12) for our methodology. In this case, we have \(\lambda \approx 1.365(p1/2)\), meaning that recovering the signal \({\varvec{x}}_{*}\) is still possible even if each measurement is flipped with probability nearly 1/2. In particular, when \(k = 1\), the measurement model (9) reduces to the traditional 1bit CS model. Therefore, the semitensorbased 1bit CS can be considered as a generalization of traditional 1bit CS, implying that Theorem 1 extends Theorem 1 in [7].
4.3 Discussion on dither technique
If \(\varvec{a}_{i}\) denotes the ith row of the measurement matrix \(\varvec{A}\), then \(q_{i}={\text{sign}}(\langle \varvec{a}_{i},{\varvec{x}}\rangle )\) indicates which side of the hyperplane \(H_{\varvec{a}_{i}}:\langle \varvec{a}_{i},\varvec{u}\rangle =0\) \((i\in [m])\) the point \({\varvec{x}}\) lies on. As all \(\varvec{a}_{i}\) are generated randomly, the sign vector \(\varvec{q}={\text{sign}}(\varvec{A}{\varvec{x}})\) encodes the cell of a random hyperplane tessellation [30] in which the signal \({\varvec{x}}\) is located. A popular strategy used for recovering \({\varvec{x}}\) is searching for a vector \({\varvec{x}}^{\#}\) that is quantization consistent, i.e., \(\varvec{q}={\text{sign}}(\varvec{A}{\varvec{x}}^{\#})\). However, due to the homogeneity of these hyperplanes, it is impossible to separate any two points on a ray from the origin even when all possible hyperplanes are utilized (see Fig.Â 1a). In other words, the measurement \(\varvec{q}={\text{sign}}(\varvec{A}{\varvec{x}})\) loses information regarding the signalâ€™s scale. Fortunately, the issue can be circumvented by introducing a dither vector prior to quantization. In the context of random tessellations, dithering can be geometrically interpreted as adding random parallel shifts to homogeneous hyperplanes, resulting in nonhomogeneous hyperplanes \(H_{\varvec{a}_{i},\tau _{i}}:\langle \varvec{a}_{i},\varvec{u}\rangle +\tau _{i}=0\) \((i\in [m])\).
As illustrated in Fig.Â 1, the random parallel shifts enable the appropriate separation between two points on a ray originating from the origin, provided that \({\varvec{\tau }}\) is suitably selected. Although we focus on unit signals in this paper, our proof process reveals that under the circumstances of semitensor product measurement, sparse signals are restored by segments. This necessitates coding the signal scale in quantization to prevent magnitude distortion during recovery. Consequently, we have employed the dithered measurement (6) and its noisy variant (9) in this study. Indeed, Chen et al. [31] proved the following result:
Proposition 2
(Lemma 1, [31]) Let x, \(\tau\) be two independent random variables satisfying \(x\le B\), \(\tau \sim uni([\gamma ,\gamma ])\) where \(\gamma \ge B\), then we have
Proposition 2 states that if all elements of \({\varvec{\tau }}\) are drawn from the uniform distribution \([\gamma ,\gamma ]\), where the bound \(\gamma >0\) is chosen such that \(\gamma \ge \Vert \varvec{A}{\varvec{x}}\Vert _{\infty }\), then the dithered measurements \({\text{sign}}(\langle \varvec{a}_{i},{\varvec{x}}\rangle +\tau _{i})\) contain most scale information on \(\langle \varvec{a}_{i},{\varvec{x}}\rangle\). By Cauchyâ€“Schwartz inequality, it is not hard to obtain
Since we assume \(\Vert {\varvec{x}}\Vert _{2}=1\) in this paper, we suggest setting \(\gamma =\max _{i\in [m]}\Vert \varvec{a}_{i}\Vert _{2}\) to generate uniformly distributed dithers \({\varvec{\tau }}\).
5 Simulations
This section initially conducts experiments on synthetic data to facilitate the selection of regularization parameters. Subsequently, we apply the proposed method to ECG signals, MRI data, and hyperspectral images (HSIs) for evaluating its effectiveness. All experiments are conducted using MATLAB (2016a) on a platform equipped with AMD Ryzen 7 5800Â H 3.20GHz CPU and 16Â GB memory.
Practical analogtodigital converters (ADCs) not only sample signals, but also quantize each measurement to a finite number of bits. The performance of ADCs is primarily limited by the quantizer, resulting in an exponential decrease in sampling rate as the resolution increases linearly [32]. This implies that higher resolution per measurement results in slower sampling rates and increased costs for the ADC. The conventional CS framework provides a solution to mitigate the quantization bottleneck by reducing the ADC sampling rate, while the 1bit CS framework directly addresses this issue in the quantization domain by decreasing the number of bits per measurement. In other words, while a sampling rate \({{\text{SR}}}>1\) is not significant in conventional compressed sensing, it may prove highly practical in 1bit (binary) systems that can acquire sign measurements at extralow bit rates [5]. We define the recovery error (RE) as \({\text{RE}}:=\Vert \hat{{\varvec{x}}}/\Vert \hat{{\varvec{x}}}\Vert _{2}{\varvec{x}}/\Vert {\varvec{x}}\Vert _{2}\Vert _{2}\), where \({\varvec{x}}\) denotes the original signal and \(\hat{{\varvec{x}}}\) denotes its estimate.
5.1 Parameter selection
Theorem 1 has indicated that the regularization parameters \(\lambda\) in (20) should be set as (25). However, determining the appropriate value for constant c remains unclear. To address this issue, we conducted a series of numerical experiments. Firstly, we generate Ndimensional signals with sparsity s randomly and evaluate the RE of the proposed method while keeping SR fixed at 20. We consider different values of c for \(N=500\), \(s=5\), \(N=1000\), \(s=10\); and \(N=2000\), \(s=20\). The average results from repeating the experiment for a hundred times are presented in Fig.Â 2a. Next, we generate 1000dimensional signals with sparsity 10 randomly and evaluate the proposed methodâ€™s REs at different SRs. In addition, we conducted 100 trials and presented the average results in Fig.Â 2b. It can be observed that \(c=0.3\) is a suitable setting for Eq.Â (20), as it yields relatively small REs across almost all cases. As the constant c does not depend on the structure of the signal, the parameter setting obtained can be applied universally. Therefore, we will maintain these settings throughout all remaining experiments in this article.
Next, we will examine the impact of factor k on recovery performance. We randomly generate signals with a length of \(N=1000\) with sparsity \(s=10\), and consider various values for k, including \(k=1\), \(k=2\), \(k=4\), \(k=5\), and \(k=10\). We repeat 100 trails and report the average REs at different SRs in Fig.Â 3. In addition to reducing the scale of the measurement matrix, we unexpectedly found from the experimental results that incorporating the semitensor product can enhance recovery accuracy. As revealed by our proof process, the semitensor productbased measurement is equivalent to the segmented measurement scheme of sparse signals. This implies that an excessively large value of k may result in measuring some zero vectors. To avoid the trivial situation, we suggest empirically selecting k such that it does not exceed the sparsity s of the signal.
5.2 ECG signal recovery
In this section, we apply the proposed method to recover a 1000length ECG signal segment randomly selected from the MITBIH Arrhythmia Database [33]. We compare the semitensor productbased measurement with other representative types of measurements in 1bit CS, including Gaussian measurement, symmetric Bernoulli measurement, and circulant Gaussian measurement. Denoted by Gaussian [7], we refer to a scenario where all entries of the measurement are independently and identically distributed according to the standard Gaussian distribution \({\mathcal {N}}(0,1)\). Denoted by SymmBernoulli [19], we mean that each element is independently sampled from a Bernoulli distribution with equal probabilities of 1 and \(1\). Referred to as CircGaussian [20], the measurement matrix was generated by shifting one element of a random row Gaussian vector to the right relative to the preceding row vector through rotation. Besides, we also compare the proposed algorithm with methods based on other sparsity regularization. For \(\ell _{0}\) regularization, we compare with a Hard Thresholdingbased method [34], denoted as HT, which aims to solve a nonconvex sparsityconstrained program. For weighted \(\ell _{1}\)norm regularization, we compare our method with the Inverse Weightsbased approach [35] (denotes as InverseWT) and the \(\ell _{1}\)norm Shannon Entropy Weightsbased method [35] (denoted as \(\ell _{1}\)SEWT).
The quantitative results are plotted in Fig.Â 4 for different sampling rates. The reconstructed results of a fraction with a length of 250, obtained at \({\text{SR}}=40\), are depicted in Fig.Â 5. The measurements based on semitensor product have consistently demonstrated superior performance in 1bit CS compared to commonly used measurements, both in terms of indicators and recovery outcomes, encompassing reconstruction performance and execution time. To evaluate the robustness of the proposed approach, we introduced random bit flips to the measurements by flipping each entry of binary measurements with a probability of 0.2. The quantitative results are plotted in Fig.Â 6 for different sampling rates, and the reconstructed results of a fraction with a length of 250 at \({\text{SR}}=40\) are illustrated in Fig.Â 7. Although the reconstructed results are slightly degraded compared to noisefree situation, our approach still demonstrates a significant advantage over its competitors. As is shown, though weighted \(\ell _{1}\)norm based methods, InverseWT and \(\ell _{1}\)SEWT, obtain better reconstruction performance than most other \(\ell _{1}\)norm based algorithms, it is still inferior to our method. We believe that our advantage primarily stems from the utilization of semitensor productbased measurements. If other sparse regularization methods are incorporated, it is possible to achieve improved recovery outcomes, which necessitates further exploration in future discussions.
5.3 MRI data recovery
We also utilize the proposed method to conduct further test on MRI data, which usually exhibit sparsity under discrete cosine transform (DCT) or wavelet basis. We used two different datasetsâ€”Data A and Data B. Data A is a cardiac cine MRI dataset distributed by the 2013 ISMRM Recon Challenge committee,^{Footnote 1} which was collected using a 2D cine breathheld bSSFP sequence with 32channel cardiac receiver coils. Data B is a publicly available myocardial perfusion MRI dataset from [36], which was acquired using a saturation recovery FLASH sequence. We introduce random bit flips to the measurements by flipping each entry of binary measurements with a probability of 0.1. To facilitate the implementation of the measurement process and enhance solving speed, we employed a blockbased sampling scheme. Empirically, we partitioned the original image into \(32\times 32\)sized parts and subsequently perform DCT on each part to obtain sparse signals through vectorization. The reconstruct is conducted in blocks. Readers concerning on further information on blockbased samplings may refer to [37, 38]. Two quantitative indicators, peak signaltonoise ratio (PSNR) and structural similarity (SSIM) [39], and execution times (in seconds) were calculated at SRs of 5, 10, 20, and 40 and are presented in TableÂ 1, where theÂ bolded data representÂ the optimal outcomes. FigureÂ 8 displays the twentieth frame along with the reconstructed results from various methods for each of the MRI datasets obtained at \({\text{SR}}=40\). It is evident, both quantitatively and visually, that our recovery method achieves better reconstruction performance. In addition, the reduced computational complexity of matrix and signal multiplication operations in semitensor productbased measurement results in significantly shorter execution time for our algorithm compared to other algorithms. This aligns with the MRIâ€™s requirement for fast sampling and reconstruction speed.
5.4 Hyperspectral images recovery
We also apply the proposed method to perform compressed sensing of HSIs. We select two typical HSI datasets, pure HYDICE Urbanpart and AVIRIS Moffett Field, both of which are sized to \(256\times 256\times 128\) for our experiments. To facilitate the implementation of the measurement process, we also employed a blockbased sampling scheme. Empirically, we partitioned the original data into \(16\times 16\times 16\)sized blocks and subsequently perform threedimensional DCT on each block to obtain sparse signals through vectorization. The recovery process is conducted in a blockwise manner. We evaluate the recovery performance by calculating the PSNR, SSIM values and execution times at SRs of 10 and 20. The quantitative indices are reported in TableÂ 2, where theÂ bolded data representÂ the optimal outcomes, and the images of all methods with bands 645155 as RGB at SR = 20 are depicted in Fig.Â 9. The superiority of both \(\ell _{1}\)SEWT and our method in terms of recovery performance is evident, with \(\ell _{1}\)SEWT exhibiting a slightly better performance than ours. However, our method exhibits a significant advantage in execution time compared to the former.
6 Conclusion
This paper proposes a model for data compression and reconstruction from onebit measurements using semitensor product. Unlike traditional measurements, this new method does not require the number of columns in the sensing matrix to be equal to the length of the underlying signal, thus breaking the dimension matching condition. Leveraging the theory of semitensor product, this approach is capable of compressing highdimensional signals with lowerdimensional sensing matrices. Theoretically, we provide an upper bound for the recovery error of this method. We also apply the proposed method to the compressed sensing of realworld data such as ECG signals, MRI data, and HSIs to show the low time complexity of our approach.
The selection of an optimal value for k may be influenced by the characteristics of the underlying signal, and this article presents empirical approaches to aid in its determination. In future research, we intend to explore a theoretical solution to this issue. Additionally, the current compressed sensing MRI method relies on a sampling matrix derived from partial Fourier transform. However, due to the limitations imposed by onebit quantization, this paper proposes the utilization of a semitensor product measurement based on Gaussian distribution, which distinguishes it from realworld scenarios. Further research is needed on how to apply 1bit quantization and semitensor productbased measurements to the real application scenarios of MRI in future. Additionally, there is still much to be investigated regarding the application of semitensor productbased measurement in highdimensional scenarios, particularly with respect to compressed sensing of lowrank matrices and tensors from binary measurements.
Availability of data and materials
Please contact any of the authors for data and materials.
References
H.F. Shen, X.H. Li, L.P. Zhang, D.C. Tao, C. Zeng, Compressed sensingbased inpainting of aqua moderate resolution imaging spectroradiometer Band 6 using adaptive spectrumweighted sparse Bayesian dictionary learning. IEEE Trans. Geosci. Remote Sens. 51(2), 894â€“906 (2014)
M. Lustig, D.L. Donoho, J.M. Santos, J.M. Pauly, Compressed sensing MRI. IEEE Signal Process. Mag. 25(2), 72â€“82 (2008)
S. Dirksen, S. Mendelson, NonGaussian hyperplane tessellations and robust onebit compressed sensing. J. Eur. Math. Soc. 23, 2913â€“2947 (2021)
P.T. Boufounos, R.G. Baraniuk, 1bit compressive sensing, in: Proceeding of the 43rd Asilomar Conference on Conference on Signals, Systems and. Computers (IEEE, Pacific Grove, 2008), pp. 16â€“21
L. Jacques, J.N. Laska, P.T. Boufounos, R.G. Baraniuk, Robust 1bit compressive sensing via binary stable embeddings of sparse vectors. IEEE Trans. Inf. Theory 59(4), 2082â€“2102 (2013)
Y. Plan, R. Vershynin, Robust 1bit compressed sensing and sparse logistic regression: a convex programming approach. IEEE Trans. Inf. Theory 59(1), 482â€“494 (2013)
L. J. Zhang, J. F. Yi, R. Jin, Efficient algorithms for robust onebit compressive sensing, in: Proceedings of the International Conference on International Conference on Machine Learning (IEEE, Beijing, 2014), pp. 820â€“828
J.Y. Hou, J.J. Wang, F. Zhang, J.W. Huang, Onebit compressed sensing via \(\ell _{p}(0<p<1)\)minimization method. Inverse Probl. 36(5), 055005 (2020)
Z. Wang, F.L. Liu, Y.H. Jia, H.Y. Yang, Y.Y. Guo, One bit compressive sensing with offgrid targets. Digit. Signal Prog. 115, 103008 (2021)
E.J. CandĂ¨s, J. Romberg, T. Tao, Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inf. Theory 52(2), 489â€“509 (2006)
R.E. Carrillo, A.B. Ramirez, G.R. Arce, K.E. Barner, B.M. Sadler, Robust compressive sensing of sparse signals: a review. EURASIP J. Adv. Signal Process. 108, 1â€“17 (2016)
J. Mo, P. Schniter, N.G. Prelcic, R.W. Heath, Channel estimation in millimeter wave MIMO systems with onebit quantization, in: Proceedings of the 48th Asilomar Conference on Signals, System and Computer (IEEE, Pacific Grove, 2014), pp. 957â€“961
Y. Noh, S. Hong, Compressed sensing based active user detection in MIMO systems with onebit ADC. IEEE Trans. Veh. Technol. 72(1), 1313â€“1317 (2022)
J.P. Xiong, Q.H. Tang, 1bit compressive data gathering for wireless sensor networks. J. Sens. 2014(7), 177â€“183 (2014)
X. Dong, Y. Zhang, A MAP approach for 1bit compressive sensing in synthetic aperture radar imaging. IEEE Geosci. Remote Sens. Lett. 12(6), 1237â€“1241 (2015)
J.Y. Hou, F. Zhang, H.Q. Qiu, J.J. Wang, Y. Wang, Y.D. Meng, Robust lowtubalrank tensor recovery from binary measurements. IEEE Trans. Pattern Anal. Mach. Intell. 44(8), 4355â€“4373 (2022)
E.J. Candes, M.B. Wakin, S.P. Boyd, Enhancing sparsity by reweighted \(\ell _{1}\) minimization. J. Fourier Anal. Appl. 14, 877â€“905 (2008)
A. Ai, A. Lapanowski, Y. Plan, R. Vershynin, Onebit compressed sensing with nonGaussian measurements. Linear Algebra Appl. 441, 222â€“239 (2014)
S. Dirksen, S. Mendelson, NonGaussian hyperplane tessellations and robust onebit compressed sensing. J. Eur. Math. Soc. 23(9), 2913â€“2947 (2021)
S. Dirksen, H.C. Jung, H. Rauhut, Onebit compressed sensing with partial Gaussian circulant matrices. Inf. Inference 9(3), 601â€“626 (2020)
Z.Q. Liu, S. Ghosh, J. Scarlett, Robust 1bit compressive sensing with partial Gaussian circulant matrices and generative priors, in: 2021 IEEE Information Theory Workshop (ITW) (IEEE, Kanazawa, 2021), pp. 1â€“6
S. Dirksen, S. Mendelson, Robust onebit compressed sensing with partial circulant matrices. Ann. Appl. Probab. 33(3), 1874â€“1903 (2023)
D.Z. Cheng, Semitensor product of matrices and its application to Morgenâ€™s problem. Sci. China Ser. 44(3), 195â€“212 (2001)
D.Z. Cheng, L.J. Zhang, On semitensor product of matrices and its applications. Acta Math. Appl. Sin. 19(2), 219â€“228 (2003)
D.Z. Cheng, H.S. Qi, A.C. Xue, A survey on semitensor product of matrices. J. Syst. Sci. Complex 20(2), 304â€“322 (2007)
D.W. Zhao, H.P. Peng, L.X. Li, S.L. Hui, Y.X. Yang, Novel way to research nonlinear feedback shift register. Sci. China Ser. 57(9), 1â€“14 (2014)
Y.H. Wu, T.L. Shen, Policy iteration approach to control residual gas fraction in IC engines under the framework of stochastic logical dynamics. IEEE Trans. Control Syst. Technol. 25, 1100â€“1107 (2017)
K.Z. Zhang, L.J. Zhang, S.S. Mou, An application of invertibility of Boolean control networks to the control of the mammalian cell cycle. IEEE ACM Trans. Comput. Biol. Bioinform. 14, 225â€“229 (2017)
D. Xie, H.P. Peng, L.X. Li, Y.X. Yang, Semitensor compressed sensing. Digit. Signal Prog. 58, 85â€“92 (2016)
Y. Plan, R. Vershynin, Dimension reduction by random hyperplane tessellations. Discrete Comput. Geom. 51(2), 438â€“461 (2014)
J.R. Chen, C.L. Wang, M.K. Ng, D. Wang, High dimensional statistical estimation under uniformly dithered onebit quantization. IEEE Trans. Inf. Theory 69(8), 5151â€“5187 (2023)
B. Le, T.W. Rondeau, J.H. Reed, C.W. Bostian, Analogtodigital converters. IEEE Signal Process. Mag. 22(6), 69â€“77 (2005)
G.B. Moody, R.G. Mark, The impact of the MITBIH arrhythmia database. IEEE Eng. Med. Biol. Mag. 20(3), 45â€“50 (2001)
J. Shen, Onebit compressed sensing via oneshot hard thresholding, in: 36th Conference on Uncertainty in Artificial Intelligence (PMLR, Toronto, 2020), pp. 510â€“519
P. Xiao, B. Zhao, Robust onebit compressive sensing with weighted \(\ell _{1}\)norm minimization. Signal Process. 164, 380â€“385 (2019)
S.G. Lingala, M. Jacob, Blind compressive sensing dynamic MRI. IEEE Trans. Med. Imaging 32(6), 1132â€“1145 (2013)
L. Gan, Block compressed sensing of natural images, in: 2007 15th International Conference on Digital Signal Processing (IEEE, Cardiff, 2007), pp. 403â€“406
J.E. Fowler, S. Mun, E.W. Tramel, Blockbased compressed sensing of images and video. Found. Trends Signal Process. 4(4), 297â€“416 (2012)
Z. Wang, A.C. Bovik, H.R. Sheikh, E.P. Simoncelli, Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600â€“612 (2004)
Acknowledgements
Not applicable.
Funding
This research was supported in part by the National Natural Science Foundation of China (Grant Nos. 12201505, 12301594), in part by the Sichuan Science and Technology Program (Grant No. 2023NSFSC0060), and in part by the Initiative Projects for Ph.D. in China West Normal University (Grant No. 22kE030).
Author information
Authors and Affiliations
Contributions
JYH proposed the framework of the whole method; performed the simulations, analysis and interpretation of the results; and drafted the manuscript. XLL has participated in the conception and revision of this research. The authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisherâ€™s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Hou, J., Liu, X. Semitensor productbased onebit compressed sensing. EURASIP J. Adv. Signal Process. 2023, 112 (2023). https://doi.org/10.1186/s13634023010716
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13634023010716