A hierarchical interference identification method based on twin-bounded support vector machine

Abstract

The application discloses a layered interference identification method based on a twin bounded support vector machine, which comprises the following steps: 1, collecting common interference signal types in a task and modeling; 2, classifying interference according to anti-interference means, and constructing an interference type layered tree; 3, decomposing a multi-classification task into multiple binary classification tasks according to the interference type layered tree; 4, modeling a channel environment, and constructing a received signal model according to a satellite communication system; 5, constructing training data for each binary classification task, and extracting low-dimensional spectrum fusion features from the received signal; 6, for each binary classification task, constructing a twin bounded support vector machine and corresponding training data set, training a twin bounded support vector machine, and obtaining two hyperplane parameters; and 7, fixing the multiple hyperplane parameters, extracting spectrum fusion features from an unknown received signal, and completing interference type identification by using the twin bounded support vector machine according to the interference type layered tree model.

Description

Technical Field

[0001] This invention relates to the field of wireless communication, and in particular to a hierarchical interference identification method based on twin bounded support vector machines. Background Technology

[0002] In satellite communications, signals are susceptible to both intentional and unintentional radio frequency interference, particularly in telemetry and tracking (TT&C) systems and global navigation satellite systems (GNSS). Intentional interference is a form of interference caused by equipment deliberately transmitting high-power signals in the communication band, while harmonics or energy leakage from other communication systems often generate unintentional interference. Because interference degrades communication performance, efficient interference suppression techniques are needed to ensure secure signals in communication links, and the design of interference type identification algorithms is fundamental to the design of interference suppression techniques. Therefore, the design of interference identification algorithms is of great significance and has attracted widespread research interest in both academia and industry.

[0003] Considering the limited computing resources of satellites, and unlike existing neural network-based identification algorithms that sacrifice computational complexity for improved accuracy, this paper investigates a fast interference identification method during data transmission, achieving good interference identification accuracy with limited computational resources. Utilizing spectrum fusion characteristics and based on prior information that certain interference types share similarities and typically employ the same anti-interference strategies, this paper proposes a novel fast interference identification algorithm. This algorithm combines Siamese bounded support vector machines and hierarchical classification, improving identification accuracy while reducing online time complexity. The proposed algorithm achieves better performance than competing methods. Summary of the Invention

[0004] To overcome the problem that satellite communication is easily affected by interference in existing technologies, this invention provides a hierarchical interference identification method based on twin bounded support vector machines, which can quickly identify the types of interference and take corresponding anti-interference measures.

[0005] To achieve the aforementioned objectives of the invention, the technical solution adopted to solve its technical problems is as follows:

[0006] A hierarchical interference identification method based on Siamese bounded support vector machines includes the following steps:

[0007] Step 1: Collect the types of common interference signals used in the task and model the common interference signals;

[0008] Step 2: Classify the types of interference based on the anti-interference methods used, and construct a hierarchical tree of interference types;

[0009] Step 3: Based on the hierarchical tree of interference types, decompose the multi-class classification into multiple binary classification tasks. Each binary classification task will be completed by a twin bounded support vector machine.

[0010] Step 4: Model the channel environment, construct the received signal model based on the satellite communication system, and generate a large number of interference-free signals and interfered signals under different interference-to-signal ratios as training data;

[0011] Step 5: Construct training data for each binary classification task, extract spectral fusion features from the received signal, and the fused spectral features consist of statistical features and dimensionality-reduced time-frequency features extracted by principal component analysis.

[0012] Step 6: For each binary classification task, construct a twin bounded support vector machine and the corresponding training dataset. Use the spectral fusion features as input data to train a twin bounded support vector machine and obtain the parameters of the two hyperplanes.

[0013] Step 7: Fix multiple hyperplane parameters, perform spectral fusion feature extraction on the unknown received signal, and use the trained twin bounded support vector machine to complete the interference type identification based on the interference type hierarchical number model.

[0014] Furthermore, step 1 includes the following:

[0015] Considering uplink satellite communication and an additive white Gaussian noise channel, the signal r(t) received by the satellite can be modeled as:

[0016]

[0017] Where s(t) and j(t) are the transmitted and jamming signals, respectively, and w(t) represents the noise. The radio frequency signal of the unified S-band USB telemetry and communication system is a cosine phase-modulated wave. On one carrier, multiple subcarriers are used for phase modulation, thereby realizing frequency division multiplexing of multiple signal transmission. Its signal can be modeled as follows:

[0018]

[0019] Where A is the carrier amplitude, w0 is the carrier angular frequency, and m i w represents the phase modulation index of the ranging tone and the subcarrier signal relative to the carrier. ci θ represents the angular frequency of the subcarrier and ranging tone signals. The subcarrier signals include telemetry, remote control, and voice signals. ci Carrier phase;

[0020] Considering common interference signals in satellite communication, there are mainly narrowband interference and wideband interference. Narrowband interference includes single-frequency interference (SFJ) and narrowband noise interference (NNJ), while wideband interference includes multi-frequency interference (MFJ), swept-frequency interference (SWFJ), and wideband noise interference (BNJ).

[0021] The single-frequency interference SFJ is a fixed continuous sine wave, which can be expressed as:

[0022]

[0023] Where, p j It is the average noise power, f j It is the carrier power, φ j ∈U(0, 2π) represents the carrier phase;

[0024] The narrowband noise interference NNJ is generated by passing Gaussian noise through a narrowband filter, and can be expressed as:

[0025]

[0026]

[0027] Where p is the average noise level, x is the noise signal, and h represents the narrowband filter;

[0028] The multi-frequency interference MFJ is a set of continuous sine waves with fixed frequencies, which can be expressed as:

[0029]

[0030] Where M represents the number of sine waves, p jm f jm φ jm These represent the average interference power, carrier frequency, and phase, respectively.

[0031] The frequency sweep interference (SWFJ) is a process of transferring full power from one frequency to another. This sweeping action rapidly and continuously blocks multiple frequencies. SWFJ can be represented as:

[0032]

[0033] Broadband noise interference (BNJ) is an important electronic countermeasure method. Broadband noise interference can be expressed as:

[0034]

[0035]

[0036] Where p represents noise power, x represents noise signal, and h represents broadband filter.

[0037] Furthermore, step 5 includes the following:

[0038] To fully utilize the hidden information in the spectrum, a core feature is constructed by fusing two components, namely:

[0039]

[0040] in, Each feature represents, in turn, the mean, variance, skewness, kurtosis, peak-to-peak value, root mean square, peak coefficient, shape coefficient, impact coefficient, edge coefficient, and energy in the frequency domain.

[0041] Secondly, considering time-frequency analysis, which can simultaneously reflect the characteristics of the time and frequency domains, the Short-Time Fourier Transform (STFT) is one of the most widely used methods in the field of time-frequency analysis. The STFT divides the observed signal window into small segments and assumes that the signal in each segment is stationary, i.e.:

[0042] R ST =STFT(r, g, q) = ∑ t r(t)h(tg)e -iwq

[0043] Where g∈[0, G-1], q∈[0, Q-1], and G and Q represent the number of analysis cycles and the number of Fast Fourier Transform (FFT) points, respectively; h(tg) represents the analysis window function, which is the Hamming function used here;

[0044]

[0045] Where vec() represents the vectorization operation;

[0046] Because r ST The dimensionality is quite high, making subsequent classification algorithms very complex. To handle high-dimensional features, Principal Component Analysis (PCA) is introduced. This method can extract r... ST The key features are:

[0047]

[0048] Among them, T PCA This represents the PCA transformation matrix.

[0049] Furthermore, step 6 includes the following:

[0050] The twin bounded support vector machine (TBSVM) algorithm is used to classify different types of interference and find a pair of non-parallel hyperplanes.

[0051]

[0052]

[0053] This causes each hyperplane to move away from data points of one class and closer to data points of another class;

[0054] Consider a binary classification problem, let x + and x- If features belong to two different classes, then empirical risk can be represented as:

[0055]

[0056]

[0057] in, Let e1 ∈ 1 represent positive and negative features respectively. m1 ,e2∈1 m2 This represents a vector whose elements are all 1s, where c1 and c2 are two positive parameters;

[0058] By introducing slack variable ξ, ξ * ,η,η * and additional regular expression terms and The binary classification problem can be transformed into the following original problem form:

[0059]

[0060]

[0061] Where d1 and d2 represent the positive penalty parameters, and the hyperplane... The inter-class single-sided spacing can be written as By minimizing the third term in equation (2), the corresponding dual problem is derived using the Lagrange method:

[0062]

[0063] Where α = [α1, ..., α] m2 ] T , β=[β1,…,β m1 ] T For a vector of Lagrange multipliers, the KKT conditions for the variables can be expressed as:

[0064]

[0065]

[0066]

[0067] -(Bw1+b1e2)+ξ2≥e2,,ξ≥0

[0068] α T (Bw1+b1e2-ξ+e2,)=0,β T ξ≥0

[0069] α≥0, β≥0

[0070] Based on the above formula, we can derive

[0071] definition E = [A e1], G = [B e2], the above formula can be expressed as:

[0072] v1 = -(E T E+d1I) -1 G T α.

[0073] Based on v1, E, and G, formula (4) can be expressed as:

[0074]

[0075] Using the KKT conditions and the expression for v1, the dual problem of equation (2) is:

[0076]

[0077] st0≤α≤c11

[0078]

[0079] st0≤η≤c21

[0080] Based on the solutions to formulas (5) and (6), the solutions to formulas (2) and (3) of the original problem can be obtained. and hyperplane

[0081] The new input feature x is classified according to which hyperplane it is closest to, i.e.:

[0082]

[0083] By employing the above technical solutions, this invention has the following advantages and positive effects compared with the prior art:

[0084] This invention can reduce the time complexity of online recognition while maintaining high interference recognition performance by leveraging interference layer prior information and machine learning techniques. Attached Figure Description

[0085] To more clearly illustrate the technical solutions of the embodiments of the present invention, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are merely some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. In the drawings:

[0086] Figure 1This is a flowchart of a hierarchical interference identification method based on twin bounded support vector machines according to the present invention;

[0087] Figure 2 This is a schematic diagram of the hierarchical tree of interference types in this invention;

[0088] Figure 3 This is a simulation diagram of the classification accuracy of the interference recognition algorithm in this invention. Detailed Implementation

[0089] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0090] This embodiment considers satellite telemetry and control uplink communication, employing an S-band telemetry and control communication system. The remote control signal is modulated by a subcarrier, and the subcarrier modulation method is BPSK. A hierarchical interference identification method based on twin bounded support vector machines is proposed for this system, such as... Figure 1-3 As shown, it includes the following steps:

[0091] Step 1: Collect the types of common interference signals used in the task and model the common interference signals;

[0092] Furthermore, step 1 includes the following:

[0093] Considering uplink satellite communication and an additive white Gaussian noise channel, the signal r(t) received by the satellite can be modeled as:

[0094]

[0095] Where s(t) and j(t) are the transmitted and jamming signals, respectively, and w(t) represents the noise. The radio frequency signal of the unified S-band USB telemetry and communication system is a cosine phase-modulated wave. On one carrier, multiple subcarriers are used for phase modulation, thereby realizing frequency division multiplexing of multiple signal transmission. Its signal can be modeled as follows:

[0096]

[0097] Where A is the carrier amplitude, w0 is the carrier angular frequency, and m i w represents the phase modulation index of the ranging tone and the subcarrier signal relative to the carrier. ci θ represents the angular frequency of the subcarrier and ranging tone signals. The subcarrier signals include telemetry, remote control, and voice signals. ci Carrier phase;

[0098] Considering common interference signals in satellite communication, there are mainly narrowband interference and wideband interference. Narrowband interference includes single-frequency interference (SFJ) and narrowband noise interference (NNJ), while wideband interference generally includes multi-frequency interference (MFJ), swept-frequency interference (SWFJ), and wideband noise interference (BNJ).

[0099] The single-frequency interference SFJ is a fixed continuous sine wave, which can be expressed as:

[0100]

[0101] Where, p j It is the average noise power, f j It is the carrier power, φ j ∈U(0, 2π) represents the carrier phase;

[0102] The narrowband noise interference NNJ is generated by passing Gaussian noise through a narrowband filter, and can be expressed as:

[0103]

[0104]

[0105] Where p is the average noise level, x is the noise signal, and h represents the narrowband filter;

[0106] The multi-frequency interference MFJ is a set of continuous sine waves with fixed frequencies, which can be expressed as:

[0107]

[0108] Where M represents the number of sine waves, p jm f jm φ jm These represent the average interference power, carrier frequency, and phase, respectively.

[0109] The frequency sweep interference (SWFJ) is a process of transferring full power from one frequency to another. This sweeping action rapidly and continuously blocks multiple frequencies. SWFJ can be represented as:

[0110]

[0111] Broadband noise jamming (BNJ) is an important electronic countermeasure method, with advantages such as simple generation and reliable masking effect, but the jamming energy is relatively large. Broadband noise jamming can be expressed as:

[0112]

[0113]

[0114] Where p represents noise power, x represents noise signal, and h represents broadband filter.

[0115] Step 2: Classify the types of interference based on the anti-interference methods used, and construct a hierarchical tree of interference types;

[0116] Taking the five types of interference mentioned above as examples, and employing different anti-interference methods based on narrowband and broadband interference, the interference type tree can be modeled as shown in the attached figure. Figure 2 form.

[0117] Step 3: Based on the hierarchical tree of interference types, decompose the multi-class classification into multiple binary classification tasks. Each binary classification task will be completed by a twin bounded support vector machine.

[0118] Step 4: Model the channel environment, construct the received signal model based on the satellite communication system, and generate a large number of interference-free signals and interfered signals under different interference-to-signal ratios as training data;

[0119] Step 5: Construct training data for each binary classification task, extract spectral fusion features from the received signal, and the fused spectral features consist of statistical features and dimensionality-reduced time-frequency features extracted by principal component analysis.

[0120] Furthermore, step 5 includes the following:

[0121] To fully utilize the hidden information in the spectrum, a core feature is constructed by fusing two components, namely:

[0122]

[0123] in, Each feature represents, in turn, the mean, variance, skewness, kurtosis, peak-to-peak value, root mean square, peak coefficient, shape coefficient, impact coefficient, edge coefficient, and energy in the frequency domain.

[0124] Secondly, considering time-frequency analysis, which can simultaneously reflect the characteristics of the time and frequency domains, the Short-Time Fourier Transform (STFT) is one of the most widely used methods in the field of time-frequency analysis. The STFT divides the observed signal window into small segments and assumes that the signal in each segment is stationary, i.e.:

[0125] R ST =STFT(r, g, q) = ∑ t r(t)h(tg)e -iwq

[0126] Where g∈[0, G-1], q∈[0, Q-1], and G and Q represent the number of analysis cycles and the number of Fast Fourier Transform (FFT) points, respectively; h(tg) represents the analysis window function, which is the Hamming function used here;

[0127]

[0128] Where vec() represents the vectorization operation;

[0129] Because r ST The dimensionality is quite high, making subsequent classification algorithms very complex. To handle high-dimensional features, Principal Component Analysis (PCA) is introduced. This method can extract r... ST The key features are:

[0130]

[0131] Among them, T PCA This represents the PCA transformation matrix.

[0132] Step 6: For each binary classification task, construct a twin bounded support vector machine and the corresponding training dataset. Use the spectral fusion features as input data to train a twin bounded support vector machine and obtain the parameters of the two hyperplanes.

[0133] Furthermore, step 6 includes the following:

[0134] The Siamese Bounded Support Vector Machine (TBSVM) algorithm is used to classify different types of interference, achieving better accuracy and time complexity than the Standard Vector Machine (SVM) algorithm. Compared to traditional support vector machines, which can only solve one large quadratic programming problem (QPP), the Siamese Bounded Support Vector Machine can solve two rigorous and complete smaller QPP problems, making it faster and more effective. Specifically, the main idea of ​​the TBSVM classifier is to find a pair of non-parallel hyperplanes:

[0135]

[0136]

[0137] This causes each hyperplane to move away from data points of one class and closer to data points of another class;

[0138] Consider a binary classification problem, let x + and x - If features belong to two different classes, then empirical risk can be represented as:

[0139]

[0140]

[0141] in, Let e1 ∈ 1 represent positive and negative features respectively. m1 ,e2∈1 m2This represents a vector whose elements are all 1s, where c1 and c2 are two positive parameters;

[0142] By introducing slack variable ξ, ξ * ,η,η * and additional regular expression terms and The binary classification problem can be transformed into the following original problem form:

[0143]

[0144]

[0145] Where d1 and d2 represent the positive penalty parameters, and the hyperplane... The inter-class single-sided spacing can be written as Optimization is achieved by minimizing the third term in equation (2). To solve these two problems, the corresponding dual problem is derived using the Lagrange method:

[0146]

[0147] Where α = [α1, ..., α] m2 ] T , β=[β1,…,β m1 ] T For a vector of Lagrange multipliers, the KKT conditions for the variables can be expressed as:

[0148]

[0149]

[0150]

[0151] -(Bw1+b1e2)+ξ2≥e2,,ξ≥0

[0152] α T (Bw1+b1e2-ξ+e2,)=0,β T ξ≥0

[0153] α≥0, β≥0

[0154] Based on the above formula, we can derive

[0155] definition E = [A e1], G = [B e2], the above formula can be expressed as:

[0156] v1 = -(E T E+d1I) -1 G T α.

[0157] Based on v1, E, and G, formula (4) can be expressed as:

[0158]

[0159] Using the KKT conditions and the expression for v1, the dual problem of equation (2) is:

[0160]

[0161] st0≤α≤c11

[0162]

[0163] s·t.0≤η≤c21

[0164] The Continuous Over-Relaxation (SOR) technique can effectively handle extremely large datasets, thus solving the two dual problems mentioned above. Based on the solutions of equations (5) and (6), the solutions to the original problems equations (2) and (3) can be obtained. and hyperplane

[0165] The new input feature x is classified according to which hyperplane it is closest to, i.e.:

[0166]

[0167] Step 7: Fix multiple hyperplane parameters, perform spectral fusion feature extraction on the unknown received signal, and use the trained twin bounded support vector machine to complete the interference type identification based on the interference type hierarchical number model.

[0168] Figure 3 This represents the recognition accuracy of different interference identification algorithms in a Gaussian additive channel under different interference-to-signal ratios, measured by the percentage of successful identification. Six types of received signals are considered during training, including interference-free signals and signals interfered with by SFJ, NNJ, MFJ, SWFJ, and BNJ. For each type of received signal, 8000 samples are used for the training set and 1000 samples for the test set. The sampling time for each sample is set to 1 ms.

[0169] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A hierarchical interference identification method based on Siamese bounded support vector machines, characterized in that, Includes the following steps: Step 1: Collect the types of common interference signals used in the task and model the common interference signals; Step 2: Classify the types of interference based on the anti-interference methods used, and construct a hierarchical tree of interference types; Step 3: Based on the hierarchical tree of interference types, decompose the multi-class classification into multiple binary classification tasks. Each binary classification task will be completed by a twin bounded support vector machine. Step 4: Model the channel environment, construct the received signal model based on the satellite communication system, and generate a large number of interference-free signals and interfered signals under different interference-to-signal ratios as training data; Step 5: Construct training data for each binary classification task, extract spectral fusion features from the received signal, and the fused spectral features consist of statistical features and dimensionality-reduced time-frequency features extracted by principal component analysis. Step 6: For each binary classification task, construct a twin bounded support vector machine and the corresponding training dataset. Use the spectral fusion features as input data to train a twin bounded support vector machine and obtain the parameters of the two hyperplanes. Step 7: Fix multiple hyperplane parameters, perform spectral fusion feature extraction on the unknown received signal, and use the trained twin bounded support vector machine to complete the interference type identification based on the interference type hierarchical number model.

2. The hierarchical interference identification method based on Siamese bounded support vector machine according to claim 1, characterized in that, Step 1 includes the following: Considering uplink satellite communication and an additive white Gaussian noise channel, the signal r(t) received by the satellite can be modeled as: Where s(t) and j(t) are the transmitted and jamming signals, respectively, and w(t) represents the noise. The radio frequency signal of the unified S-band USB telemetry and communication system is a cosine phase-modulated wave. On one carrier, multiple subcarriers are used for phase modulation, thereby realizing frequency division multiplexing of multiple signal transmission. Its signal can be modeled as follows: Where A is the carrier amplitude, w0 is the carrier angular frequency, and m i w represents the phase modulation index of the ranging tone and the subcarrier signal relative to the carrier. ci θ represents the angular frequency of the subcarrier and ranging tone signals. The subcarrier signals include telemetry, remote control, and voice signals. ci Carrier phase; Considering common interference signals in satellite communications, there are narrowband interference and wideband interference. Narrowband interference includes single-frequency interference (SFJ) and narrowband noise interference (NNJ), while wideband interference includes multi-frequency interference (MFJ), swept-frequency interference (SWFJ), and wideband noise interference (BNJ). The single-frequency interference SFJ is a fixed continuous sine wave, which can be expressed as: Where, p j It is the average noise power, f j It is the carrier power, φ j ∈U(0, 2π) represents the carrier phase; The narrowband noise interference NNJ is generated by passing Gaussian noise through a narrowband filter, and can be expressed as: Where, p j Let x be the average noise power, h be the noise signal, and h be the noise signal. n Indicates a narrowband filter; The multi-frequency interference MFJ is a set of continuous sine waves with fixed frequencies, which can be expressed as: Where M represents the number of sine waves, p jm f jm φ jm These represent the average interference power, carrier frequency, and phase, respectively. The frequency sweep interference (SWFJ) is a process of transferring full power from one frequency to another. This sweeping action rapidly and continuously blocks multiple frequencies. SWFJ can be represented as: Broadband noise interference (BNJ) is an important electronic countermeasure method. Broadband noise interference can be expressed as: Where p represents noise power, x represents noise signal, and h w This represents a broadband filter.

3. The hierarchical interference identification method based on Siamese bounded support vector machine according to claim 1, characterized in that, Step 5 includes the following: To fully utilize the hidden information in the spectrum, a core feature is constructed by fusing two components, namely: in, Each feature represents, in turn, the mean, variance, skewness, kurtosis, peak-to-peak value, root mean square, peak coefficient, shape coefficient, impact coefficient, edge coefficient, and energy in the frequency domain. Secondly, considering time-frequency analysis, which can simultaneously reflect the characteristics of the time and frequency domains, the Short Time Fourier Transform (STFT) divides the observed signal window into small segments and assumes that the signal in each segment is stationary, i.e.: R ST =STFT(r,g,q)=∑ t r(t)h(t-g)e -iwq Where g∈[0, G-1], q∈[0, Q-1], and G and Q represent the number of analysis cycles and the number of Fast Fourier Transform (FFT) points, respectively; h(t - g) represents the analysis window function, here we take the Hamming function; Where vec() represents the vectorization operation; Principal component analysis (PCA) was introduced, which can extract r ST The key features are: Among them, T PCA This represents the PCA transformation matrix.

4. The hierarchical interference identification method based on Siamese bounded support vector machine according to claim 1, characterized in that, Step 6 includes the following: The twin bounded support vector machine (TBSVM) algorithm is used to classify different types of interference and find a pair of non-parallel hyperplanes. This causes each hyperplane to move away from data points of one class and closer to data points of another class; Consider a binary classification problem, let x + and x - If features belong to two different classes, then empirical risk can be represented as: in, Let e1 ∈ 1 represent positive and negative features respectively. m1 ,e2∈1 m2 This represents a vector whose elements are all 1s, where c1 and c2 are two positive parameters; By introducing slack variable ξ, ξ * ,η,η * and additional regular expression terms and The binary classification problem can be transformed into the following original problem form: stAw1+b1e1=ξ * ,-(Bw1+b1e2)+ξ≥e2,ξ≥0 s.t.Bw2+b2e2=η * ,(Aw2+b2e1)+η≥e1,η≥0 Where d1 and d2 represent the positive penalty parameters, and the hyperplane The inter-class single-sided spacing can be written as By minimizing the third term in equation (2), the corresponding dual problem is derived using the Lagrange method: Where α=[α1,…,α m2 ] T ,β=[β1,…,β m1 ] T For a vector of Lagrange multipliers, the KKT conditions for the variables can be expressed as: -(Bw1+b1e2)+ξ2≥e 2, ,ξ≥0 a T (Bw1+b1e2-ξ+e 2, )=0,β T ξ≥0 α≥0,β≥0 Based on the above formula, we can derive definition E = [A e1], G = [B e2], the above formula can be expressed as: v1=-(E T E+d1I) -1 G T α. Based on v1, E, and G, formula (4) can be expressed as: Using the KKT conditions and the expression for v1, the dual problem of equation (2) is: st 0≤α≤c11 st 0≤η≤c21 Based on the solutions to formulas (5) and (6), the solutions to formulas (2) and (3) of the original problem can be obtained. and hyperplane The new input feature x is classified according to which hyperplane it is closest to, i.e.:

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