Quantum Relief Algorithm

Machine learning refers to an area of computer science in which patterns are derived (“learned”) from data with the goal to make sense of previously unknown inputs. As part of both artificial intelligence and statistics, machine learning algorithms process large amounts of information for tasks that come naturally to the human brain, such as image recognition, pattern identification and strategy optimization.

Machine learning tasks are typically classified into two broad categories 1 , supervised and unsupervised machine learning, depending on whether there is a learning “signal” or “feedback” available to a learning system. In supervised machine learning, the learner is provided a set of training examples with features presented in the form of high-dimensional vectors and with corresponding labels to mark its category. On the contrary, no labels are given to the learning algorithm, leaving it on its own to find structure in unsupervised machine learning. The core mathematical task for both supervised and unsupervised machine learning algorithm is to evaluate the distance and inner products between the high-dimensional vectors, which requires a time proportional to the size of the vectors on classical computers. As we know, it is “curse of dimensionality” 2 to calculate the distance in high-dimensional condition. One of the possible solutions is the dimension reduction 3 , and the other is the feature selection 4 5 .

Relief algorithm is one of the most representative feature selection algorithms, which was firstly proposed by Kira et al. 6 . The algorithm devises a “relevant statistic” to weigh the importance of the feature which has been widely applied in many fields, such as, hand gesture recognition 7 , electricity price forecasting 8 and power system transient stability assessment 9 . Relief algorithm will occupy larger amount of computation resources while the number of samples and features become huger, which restricts the application of the algorithm.

Since the concept of quantum computer was proposed by the famous physicist Feynman 10 , a number of remarkable outcomes have been proposed. For example, Deutsch’s algorithms 11 12 embody the superiority of quantum parallelism calculation, Shor’s algorithm 13 solves the problem of integer factorization in polynomial time, and Grover’s algorithm 14 has a quadratic speedup to the problem of conducting a search through some unstructured search space. With the properties of superposition and entanglement, quantum computation has the potential advantage for dealing with high dimension vectors which attract researchers to apply the quantum mechanics to solve some classical machine learning tasks such as quantum pattern matching 15 , quantum probably approximately correct learning 16 , feedback learning for quantum measurement 17 , quantum binary classifiers 18 19 , and quantum support vector machines 20 .

Even in quantum machine learning, we still face with the trouble of “curse of dimensionality”, thus dimension reduction or feature selection is a necessary preliminary before training high-dimensional samples. Inspired by the idea of computing the inner product between two vectors 21 22 , we propose a Relief-based quantum parallel algorithm, also named quantum Relief algorithm, to effectively perform the feature selection.

The outline of this paper is as follows. The classic Relief algorithm is briefly reviewed in Sect. 2, the proposed quantum Relief algorithm is proposed in detail in Sect. 3, and a simulation experiment based on IBM Q with a simple example is given in Sect. 4. Subsequently, the efficiency of the algorithm is analyzed in Sect. 5, and the brief conclusion and the outlook of our work are discussed in the last section.

Relief algorithm is a feature selection algorithm used in binary classification (generalizable to polynomial classification by decomposition into a number of binary problems) proposed by Kira and Rendell 6 in 1992. It is efficient in estimating features according to how well their values distinguish among samples that are near each other.

We can divide an $M$-sample set into two vector sets: $A{\rm{=}}\left\{{{v_{j}}{\rm{|}}{v_{j}}\in{{\mathbb{R}}^{\rm{N}}},j=1,2,\cdots{M_{1}}}\right\}$, $B{\rm{=}}\left\{{{w_{k}}{\rm{|}}{w_{k}}\in{{\mathbb{R}}^{\rm{N}}},k=1,2,\cdots{M_{2}}}\right\}$, where ${v_{j}}$, ${w_{k}}$ are $N$-feature samples: ${v_{j}}={\left({{v_{j1}},{v_{j2}},\cdots,{v_{jN}}}\right)^{\mathsf{T}}}$, ${w_{k}}={\left({{w_{k1}},{w_{k2}},\cdots{w_{kN}}}\right)^{\mathsf{T}}}$, ${v_{j1}},\cdots,{v_{jN}},{w_{k1}},\cdots,{w_{kN}}\in\{0,1\}$, and the weight vector of $N$ features $WT={\left({w{t_{1}},w{t_{2}},\cdots,w{t_{N}}}\right)^{\mathsf{T}}}$ is initialized to all zeros. Suppose the upper limit of iteration is $T$, and the relevance threshold (that differentiate the relevant and irrelevant features) is $\tau(0\leq\tau\leq 1)$. The process of Relief algorithm is described in Algorithm 1.

At each iteration, pick a random sample $u$, and then select the samples closest to $u$ (by $N$-dimensional Euclidean distance) from each class. The closest same-class sample is called Near-hit, and the closest different-class sample is called Near-miss. Update the weight vector such that,

where the function $\emph{diff}(i,u,v)$ is defined as below,

Relief was also described as generalizable to polynomial classification by decomposition into a number of binary problems. However, as the scale of samples and the number of features increase, their efficiencies will drastically decline.

In order to handle the problem of large samples and large features, we propose a quantum Relief algorithm. Suppose the sample sets $A=\left\{{{v_{j}},j=1,2,\cdots{M_{1}}}\right\}$, $B=\left\{{{w_{k}},k=1,2,\cdots{M_{2}}}\right\}$, weight vector $WT$, the upper limit $T$ and threshold $\tau$ are the same as classical Relief algorithm defined in Sec. 2. Different from the classical one, all the features of each sample are represented as a quantum superposition state, and the similarity between two samples can be calculated in parallel. Algorithm 2 shows the detailed procedure of quantum Relief algorithm.

Suppose there are four samples(see Tab. 1), ${S_{0}}=(1,0,1,0)$, ${S_{1}}=(1,0,0,0)$, ${S_{2}}=(0,1,1,0)$, ${S_{3}}=(0,1,0,0)$, thus the $n$ in Eq. (4) is 2, and they belong to two classes: $A=\{{S_{0}},{S_{1}}\}$, $B=\{{S_{2}},{S_{3}}\}$, which is illustrated in Fig. 4.

First, the four initial quantum states are prepared as follows,

Taking ${\left|\psi\right\rangle_{{{\rm{S}}_{\rm{0}}}}}$ as an example, the ${H^{\otimes 2}}$ operation is applied on the third and fourth qubits,

Then we perform ${R_{y}}$ rotation (see Eq. (6)) on the last qubit, where

and can get

The other quantum states are prepared in the same way and they are listed as below,

Second, we randomly select a sample (assume ${\left|\phi\right\rangle_{{{\rm{S}}_{\rm{0}}}}}$ is that one), and perform similarity calculation with other samples (i.e., ${\left|\phi\right\rangle_{{{\rm{S}}_{\rm{1}}}}}$, ${\left|\phi\right\rangle_{{{\rm{S}}_{\rm{2}}}}}$, ${\left|\phi\right\rangle_{{{\rm{S}}_{\rm{3}}}}}$). Taking ${\left|\phi\right\rangle_{{{\rm{S}}_{\rm{0}}}}}$ and ${\left|\phi\right\rangle_{{{\rm{S}}_{\rm{1}}}}}$ as an example, we perform a swap operation between the last two qubits of ${\left|\phi\right\rangle_{{{\rm{S}}_{\rm{0}}}}}$,

After that, the swap test operation is applied on ($\left|{{\varphi}}\right\rangle$, $\left|{{\phi}}\right\rangle_{{{\rm{S}}_{\rm{1}}}}$),

then, we measure the result shown in Eq.(24) and obtain the probability of the first qubit being $\left|{\rm{1}}\right\rangle$ is

In terms of Eq. (12), the inner product between $\left|{{\varphi}}\right\rangle$ and $\left|{{\phi}}\right\rangle_{S_{1}}$ can be represented as

From Eqs. (25) and (26), the similarity between $\left|{{S_{0}}}\right\rangle$ and $\left|{{S_{1}}}\right\rangle$ is

here, the value of ${P_{1}^{0}}(1)$ can be determined by measurement.

In order to obtain the measurement result and also verify our algorithm, we choose the IBM Q [24] to perform the quantum processing (Fig. 5 gives the schematic diagram of our experimental circuit)¹¹1In the experiment, we program our algorithm based on the QISKit toolkit[25] and phyton language, and remotely connect the online IBM QX5 device to execute quantum processing.. After the experiment, we can get ${P_{1}^{0}}(1)$ which is shown in Tab. 2.

According to ${P_{1}^{0}}(1)$ (from Tab. 2) and Eq. (27), we can calculate the similarity between $\left|{{S_{0}}}\right\rangle$ and ${\left|S_{1}\right\rangle}$,

In the same way, the other two similarities ($\left|{{S_{0}}}\right\rangle$, ${\left|S_{2}\right\rangle}$), ($\left|{{S_{0}}}\right\rangle$, ${\left|S_{3}\right\rangle}$) can also be obtained (which are illustrated in Tab. 3).

Third, From Tab. 3, it is easy to find Near-hit is $S_{1}$ and Near-miss is $S_{3}$ (as shown in Fig. 6). Then, the weight vector is updated by applying Eq.(15), and the result of $WT$ is listed in the second row of Tab. 4 for the first iteration.

The algorithm iterates $T$ times (in our example, $T$=4) as above steps, and obtains all the $WT$ results shown in Tab. 4. After $T$-th iterations, $WT=[4,4,-2,0]$, then $\overline{WT}=[1,1,-1/2,0]$. Since the threshold $\tau=0.5$, so the selected features are $F_{0}$ and $F_{1}$, i.e., the first and second features.

In classical Relief algorithm, it takes O(N) times to calculate the Euclidean distance between $u$ and each of the $M$ samples, the complexity of finding its Near-hit and Near-miss is related to $M$, and the loop iterates $T$ times, so the computational complexity of classical Relief algorithm is O(TMN). Since $T$ is a constant which affects the accuracy of relevance levels, but it is chosen independently of $M$ and $N$, so the complexity can be simplified to O(MN). In addition, an $N$-dimensional vector in the Hilbert space is represented by $N$ bits in the classical computer, and there are $M$ samples (i.e., $M$ $N$-dimensional vectors) needed to be stored in the algorithm, so the classical Relief algorithm will consume $O(MN)$ bits storage resources.

In our quantum Relief algorithm, all the features of each sample are superposed on a quantum state ${\left|{{\phi_{A}}}\right\rangle_{j}}$ or ${\left|{{\phi_{B}}}\right\rangle_{k}}$, then the similarity calculation between two states, which is shown in Eq. (13), is just required to be taken $O(1)$ time. As same as Relief algorithm, the similarity between the selected state $\left|\varphi\right\rangle$ and each state in $\left\{{\left|{{\phi_{A}}}\right\rangle}\right\}$, $\left\{{\left|{{\phi_{B}}}\right\rangle}\right\}$ is calculated, taking $O(M)$ times, to obtain Near-miss and Near-hit, and the loop iterates $T$ times, so the proposed algorithm totally takes O(TM) times. Since $T$ is a constant, the computational complexity can be rewritten as O(M). On the viewpoint of resource consumption, each quantum state in state sets $\left\{{\left|{{\phi_{A}}}\right\rangle_{j}}\right\}$ and $\left\{{\left|{{\phi_{B}}}\right\rangle_{k}}\right\}$ is represented as

or

and it consists of $O({\log_{2}}N)$ qubits. Since $j=1,2,\cdots,{M_{1}}$, $k=1,2,\cdots,{M_{2}}$ and $M={M_{1}}+{M_{2}}$, that means that there are such $M$ quantum states needed to be stored in our algorithm, so it will consume $O(MlogN)$ qubits storage resources.

Tab. 5 illustrates the efficiency comparison, including the computational complexity and resource consumption, between classical Relief and our quantum Relief algorithms. As shown in the table, the computational complexity of our algorithm is $O(M)$, which is obviously superior than $O(MN)$ in classical Relief algorithm. On the other hand, the resource that our algorithm needs to consume is $O(MlogN)$ qubits, while the classical Relief algorithm consumes $O(MN)$ bits.

With quantum computing technologies nearing the era of commercialization and quantum supremacy, recently machine learning seems to be considered as one of the ”killer” applications. In this study, we utilize quantum computing technologies to solve a simple feature selection problem (just used in binary classification), and propose the quantum Relief algorithm, which consist of four phases: state preparation, similarity calculation, weight vector update, and features selection. Furthermore, we verify our algorithm by performing quantum experiments on the IBM Q platform. Compared with the classical Relief algorithm, our algorithm holds lower computation complexity and less resource consumption (in terms of number of resource). To be specific, the complexity is reduced from $O(MN)$ to $O(M)$, and the consumed resource is shortened from $O(MN)$ bits to $O(MlogN)$ qubits.

Although this work just focuses on the feature selection problem (even the simple binary classification), but the method can be generalized to implement the other Relief-like algorithms, such as ReliefF 26 , RReliefF 27 . Besides, we are interested in utilizing quantum technologies to deal with some classic high-dimension massive data processing, such as text classification, video stream processing, data mining, computer version, information retrieval, and bioinformatics.