NMF is to find a non-negative base-matrix and a coefficient-matrix, and it meet equation of R≈WH . It makes all components after decomposition non-negative, and at the same time achieves a non-linear dimension reduction [30–32]. It has good explanation of the local characteristics of things and it can be used to discover image features in the database, to facilitate rapid and automatic identification of applications; to discover the semantic relevance of documents for automatic indexing and extraction of information; to identify genes in DNA array analysis, and so on [33–36].

The raw matrix Rn×m is decomposed into the product of two matrices Wn×t and Ht×m :

(1) R≈WH=∑a=1tWiaHaμ

where the choice of t value usually satisfied (n+m)t<nm , the raw matrix R is compressed to WH . Each column in the R matrix represents an observation, and each row represents a feature. By replacing the raw matrix with the coefficient matrix H , it is possible to obtain a dimensionality reduction matrix of the data features, thereby reducing the storage space [37–39].

The literature [40–43] pointed out that kernel method mapping data by a kernel function. New space, then use non-negative matrix factorization for features extraction and analysis in new spaces. On the other hand, kernel method does not require knowledge of the details of the data, only the connection between each data, and finally, the kernel method can handle those data points with negative characteristics [44,45].

Kernel method firstly performs data mapping:

(2) ϕ:x∈R→ϕ(x)∈ϕ(R)

Then do the normal NMF in the new space:

(3) ϕ(R)=WϕH

Because it does not know the specific mapping function, the paper need to use the kernel method to get:

(4) (ϕ(R))Tϕ(R)=(ϕ(R))TWϕH

It is easy to see that the left side of the above equation is the definition of the kernel. The paper does not need to know the detailed mapping definitions, just need to know their inner product. The choice of kernel can make Gaussian kernel, polynomial kernel and so on. By replacing the above formula with a nuclear function, it can get:

(5) K=YH

where, K=(ϕ(R))Tϕ(R) , Y=(ϕ(R))TWϕ

Kernel function is to map data from low-dimensional to a new space, to facilitate the conversion of data that cannot be linearly segmented into data that can be linearly segmented, usually can be described as:

(6) F(a,b)=⟨ϕ(a),ϕ(b)⟩=(ϕ(a))Tϕ(b)

It can be seen that the equation and the ordinary non-negative matrix factorization have the same representation, so the same method can be used for solving.

Given a new matrix Rnew , it can get a new coefficient matrix:

(7) Hnew=(Wϕ)−1ϕ(Rnew)=(Wϕ)−1((ϕ(R))T)−1(ϕ(Rnew))Tϕ(Rnew)=Y−1Knew

where, Y−1=(Wϕ)−1((ϕ(R))T)−1 , Knew=(ϕ(R))Tϕ(Rnew) .

The flow diagram of privacy-preserving process of user data is shown in Fig. 1. Privacy-preserving hides raw data through random perturbation techniques. The specific steps are as follows:

First, random perturbation techniques are used in the privacy-preserving of mining. An intuitive method is to add a number of d , so that a+d appears in the database instead of the raw data a , where d is a random number obeying a certain distribution. Although we can do not specific operations on a single data a , if we care about the entire database rather than a single data, then we can still perform the corresponding operations. The method is to randomly perturb the raw data, so that the database in the server saves the changed data instead of the raw data, thereby achieving the purpose of hiding information and privacy-preserving.

Given a recommendation system with n items, m users, for reasons of privacy-preserving, the side of server do not know the raw rating of each user and items the user rated. To do this, it uses a uniform distribution to generate random data. In a uniform distribution, all users produce a random value in the interval [μ−α,μ+α] . In a normal distribution, all users produce a normal distribution random value and it combine the mean of μ with the standard deviation of α . The user hides the rating data before sending its rating data to the server. The specific data hiding steps are as follows.

The server determines the specific distribution (uniform distribution or normal distribution) of the disturbance data and the corresponding parameters (α,μ,σ) and informs each user of this information.

Each user fills the unsorted items with the mean of their existing ratings.

For each user u use a certain distribution to generate n random numbers, i=1,2,...,n . Then the user data adds a random number to get a vector of hidden information:

(8) zui′=zui+dui,  i=1,2,…,n

Finally, the server uses the Eq. (8) to form a rating matrix Rʹ of hidden information.

The flow diagram of the recommendation process of user data is shown in the Fig. 2. The feature matrix is obtained by matrix decomposition of the hidden data obtained above, and then the user-related data is calculated by using the new user data and the feature matrix, and the neighbors are found by the similarity, and finally weighted to obtain the recommended data. Specific steps are as follows.

The steps of NMF for privacy-preserving are as follows:

Kernel method has improved the basic idea of NMF. The privacy-preserving step is similar to the NMF, and finally a rating matrix of hidden information is obtained. The kernel method of the hidden matrix is decomposed as follows.

This algorithm consists of two parts: offline data preparation and online data processing. The first part is offline data preparation. It mainly does the nonnegative matrix decomposition of the hidden matrix. Because this part is offline, it is prepared for the second part in advance, which does not affect the online recommendation generation, so the time complexity of this part will not be considered. The second part includes the calculation of Hnew , the calculation of Euclidean distance, neighbors search and recommendation generation. The time complexity of Hnew calculation is O(t∗m) , the time complexity of Euclidean distance is O(t∗n) , the time complexity of neighbors search is O(n∗log⁡n) , and the time complexity of recommendation generation is O(k∗n) . So, the time complexity of the second part is O(t∗m)+O(t∗n)+O(n∗log⁡n)+O(k∗n) . Since k and t are constants, m and n have the same order of magnitude, the time complexity of the second part is O(n∗log⁡n) . In conclusion, the time complexity of the algorithm is O(n∗log⁡n) .

MovieLens is the oldest recommendation system. The MovieLens dataset contains rating data for multiple movies from multiple users, as well as movie data information and user attribute information. This data set is often used as a test data set for recommendation systems, machine learning algorithms. The content in the file contains the rating of each user for each movie. The data in our experiments consists of 1,000,000 ratings for 6,040 users with 3,952 movies, and all users must rate at least 20 movies.

The mean absolute deviation is a statistic that describes the degree of data dispersion. The mean absolute error (MAE) is:

(12) MAE=∑Ni=1|pi−qi|N

where, pi indicates the prediction rating of the recommendation system for the i item, qi indicates real ratings of the i items, and N indicates the number of items.

It can be known from the Eq. (12) that the accuracy of the prediction is represented by calculating the MAE value between the predicted rating and the real ratings.

This paper demonstrates the effectiveness of the proposed algorithm by comparing the kernel method privacy-preserving algorithm with the non-negative matrix factorization privacy-preserving algorithm. At first, find the mean of the user’s rated movie to fill the corresponding unrated movie. Then, each user creates a new privacy-preserving data rating matrix by creating N random values using a Gaussian distribution to hide the raw data.

This experiment mainly verifies the performance of the recommendation system by predicting the rating of the known movie and comparing the real rating of the movie with the predicted rating. Basically, it can be understood that the rating value of the predicted movie is set to null, and the prediction rating is performed by using the hidden matrix and the kernel method-based privacy-preserving algorithm proposed in this paper. Since the matrix decomposition results close to the hidden matrix each time, but the results of each decompose are different, this paper performs 20 matrix decomposition to narrow the difference of prediction results, and finally takes the mean value of the prediction results as the movie rating prediction value, it can narrows the difference of the results.

First, this paper examines the influence of the value of t about the matrix decomposition on the prediction accuracy, and the t value of 5 to 15. Experiments are performed on 6040 users and 3952 projects using non-negative matrix factorization and kernel method respectively to examine the effect of t-value on prediction accuracy. The result is shown in Fig. 3.

In Fig. 3, with the increasing of the value of t after decomposition, the value of MAE shows a steady trend, indicating that the number of t has no significant influence on the prediction accuracy, and is basically in a flat state. The reason should be that when the user or project data is sufficiently rich, the sample mean and variance of the perturbed data will converge to their expected μ . Therefore, the value of users or projects are relatively rich, the system prediction accuracy has no significant impact. In general, due to the using of hidden information for prediction, the accuracy of non-negative matrix factorization and kernel method is slightly lower than that of the direct use of the original score data. The error between the interference and the pre-interference is small, and certain recommendation accuracy can be achieved.

In Fig. 3, when the value of t after decomposition is 6 and 9, the prediction accuracy is high. The following is a comparison of the MAE cases with different neighbors k when the value of t after decomposition is 6 and 9, respectively.

When t after decomposition is 6, the value of MAE of different neighbor k . As shown in Fig. 4.

When t after decomposition is 9, the value of MAE of different neighbor k . As shown in Fig. 5.

Because non-negative matrix factorization and kernel method predictions mainly use the similarity between new users and existing scoring users to make recommendations, the impact of the number of neighbors on prediction accuracy is investigated. Using the data set experiment of 6040 by 3952, the number of neighbors is changed separately, and the corresponding prediction result is obtained. The results of experiment are shown in Fig. 6.

In Fig. 6, with the increasing of the value of neighbors k , the value of MAE shows a steady trend, indicating that the prediction accuracy is not sensitive to k . The reason should be that when the user or project data is sufficiently rich, that is, the data is relatively rich, and the sample mean and variance of the disturbance data will converge to their desired μ . Therefore, as the number of k increases, the prediction accuracy of the system tends to be stable. It can be proved that the kernel method is slightly higher than non-negative matrix factorization. Kernel method maps data to a new space, and the data range is reduced, and the corresponding error will also be smaller. Therefore, compared to non-negative matrix factorization, the predictive performance of the kernel method is relatively better.

In Fig. 6, when the value of k is 9, 10, the prediction accuracy is high. The following compares the prediction accuracy of the value of t after different decompositions when k is 9, 10, respectively.

When k is 9, the value of MAE of t is different after decomposition. As shown in Fig. 7.

When k is 10, the value of MAE of t is different after decomposition. As shown in Fig. 8.

In order to examine the influence of the dispersion degree of the disturbance data on the prediction accuracy, a 6040 by 3952 data set experiment was used to change the variance of the disturbance data and examine the results of the corresponding prediction. The results of experiment are shown in Fig. 9.

Obviously, the degree of interference of raw data has a greater impact on the prediction. When the interference degree of the raw data is small, the accuracy of the prediction result of the recommended system is better.

From above experiments, the results of the perturbation data using kernel method are slightly better than non-negative matrix factorization. Moreover, from the previous validity analysis, no matter what kind of disturbance distribution, the mean absolute error using the same algorithm should converge to the error of the undisturbed data in the large samples.

This paper proposed a kernel method-based privacy-preserving collaborative filtering algorithm that is easy to implement and can guarantee effectiveness of recommendation. This algorithm mainly involves two parameters, one is the dimension value t in matrix decomposition, the other is the number of neighbors k of the current user. The experimental results on page 8–9 show that the algorithm is relatively insensitive to t value, and small changes have little impact on the results; the experimental results on page 9–10 show that the algorithm is also insensitive to k value, and the small changes of its value can be ignored. To sum up, the settings of these two parameters can be adjusted according to the actual application scenarios. Compared to the NMF-based privacy-preserving algorithm, its corresponding prediction accuracy has been improved. And compared to the direct recommendation on non-hidden data, although the accuracy is reduced, the performance is acceptable as a compromise of privacy-preserving. The limitation of this algorithm is that when the number of users and items is large, the time required for NMF or KNMF matrix decomposition is relatively long. However, matrix decomposition can be carried out offline, it has little effect on online user neighbors search and recommendation generation.