CN104318046A - System and method for incrementally converting high dimensional data into low dimensional data - Google Patents

Abstract

The invention discloses a system and a method for incrementally converting high dimensional data into low dimensional data. The system comprises a high dimensional data acquisition system, wherein the high dimensional data acquisition system is connected with a data processing system, the data processing system comprises a module used for incrementally converting the high dimensional data into the low dimensional data, and the data processing system comprises a queue used for storing the high dimensional data. The structure is combined with the method of the structure to avoid defects in the prior art that the hardware processing effect of the data processing system is low, time and labor are wasted, a processing process is slow due to crash under severe situations on an aspect of concurrent execution processing, and data is lost and even real-time state information can not be normally reflected under the environment of the real-time processing of the high-dimensional data.

Description

The high dimensional data of increment type is converted to the system and method for low-dimensional data

Technical field

The invention belongs to high dimensional data and the processing technology field of increment type, the high dimensional data being specifically related to a kind of increment type is converted to the system and method for low-dimensional data.

Background technology

Existing in scientific research and engineer applied, much by data acquisition system come as video, audio frequency, weather and view data have the feature of high dimensional data, this high dimensional data can provide abundant and detailed information, but the process of high dimensional data often produces the processing time of the excessive magnanimity caused of dimension, this problem often causes the treatment effeciency effect of the hardware of data handling system not high, to take time and effort and meeting serious in concurrence performance process causes deadlock to neglect treatment progress, if under the environment processing high dimensional data in real time, this will cause the loss of data and even normally cannot reflect real time status information.

Summary of the invention

Object of the present invention provides a kind of high dimensional data of increment type to be converted to the system and method for low-dimensional data, comprise high dimensional data acquisition system, described high dimensional data acquisition system is connected with data handling system, the high dimensional data included in described data handling system for increment type is converted to the module of low-dimensional data, includes the queue for depositing high dimensional data in described data handling system.If such structure avoid in prior art in conjunction with its method cause the treatment effeciency effect of the hardware of data handling system not high, to take time and effort and meeting serious in concurrence performance process causes crashing and neglects treatment progress and this defect that will cause the loss of data and even normally cannot reflect real time status information under the environment of real-time process high dimensional data.

In order to overcome deficiency of the prior art, the high dimensional data that the invention provides a kind of increment type is converted to the solution of the system and method for low-dimensional data, specific as follows:

A kind of high dimensional data of increment type is converted to the system of low-dimensional data, comprise high dimensional data acquisition system 1, described high dimensional data acquisition system 1 is connected with data handling system 2, the high dimensional data included in described data handling system 2 for increment type is converted to the module 3 of low-dimensional data, includes the queue 4 for depositing high dimensional data in described data handling system 2.

The high dimensional data of described increment type is converted to the method for the system of low-dimensional data, as follows:

Step 1: first high dimensional data acquisition system carries out the collection for the such high dimensional data of video, audio frequency, weather or view data, is then sent to data handling system 2 by the high dimensional data collected;

Step 2: after data handling system 2 receives high dimensional data, then according to the sequencing received, high dimensional data is stored in for depositing in the queue 4 of high dimensional data successively, start and be used for the module 3 that the high dimensional data of increment type is converted to low-dimensional data and set a n-dimensional space object V, include k dimension space object S in described n-dimensional space object V, k is initially set to 0;

Step 3: then data handling system 2 takes out a high dimensional data from the queue 4 for depositing high dimensional data sequentially successively, after taking out a high dimensional data X, just carry out the extraction to the characteristic component of this high dimensional data X and dimensionality reduction operation, described high dimensional data X is expressed as (x ₁, x ₂... x _n), n is the dimension of this high dimensional data;

Step 4: described carrying out comprises to the extraction of the characteristic component of this high dimensional data and dimensionality reduction operation the module 3 being first converted to low-dimensional data for the high dimensional data of increment type and this high dimensional data X is projected in the k dimension space represented by k dimension space object S, and this high dimensional data X method projected in the k dimension space S represented by k dimension space object obtains result vector r according to formula (1) with alternative manner _k:

$r_k=X-\underset{i=1}{\overset{k}{\mathrm{\Σ}}}{b}_i{b}_i^{T}X---\left(1\right)$

The span of described i is be the dimension of current spatial object S from 1 to k, k, is also the dimension of low-dimensional after current Data Dimensionality Reduction, the first coefficient r _kfor result vector, when the length of result vector || r _k|| ₂be less than T _ktime, k value is constant, and current spatial object S also remains unchanged, || r _k|| ₂be more than or equal to T _ktime, try to achieve kth+1 coefficient by b _k+1add former k dimension space object S as new spatial base, make the dimension of spatial object S increase by 1, k=k+1.Setting r ₀=X, and r _ifor intermediate vector, T _ibe the i-th threshold value, x _maxfor that high dimensional data that data length in the queue 4 for depositing high dimensional data is the longest;

Step 5: according to obtained b ₁, b ₂b _kthe part Y of component non-zero in the dimensionality reduction data vector obtaining for this high dimensional data X by formula (2):

$Y=\left[\begin{array}{c}{X}^T{b}_1\\ X^T{b}_2\\ ...\\ X^T{b}_i\\ ....\\ X^T{b}_k\end{array}\right]---\left(2\right)$

After the queue of high dimensional data is all disposed, according to the value of final k, the part of the dimensionality reduction data vector component non-zero of whole high dimensional data is supplemented the part that upper component is zero, be collectively expressed as k dimension dimensionality reduction after data vector.

Step 6: after obtaining the dimensionality reduction data vector for each high dimensional data, if high dimensional data acquisition system proceeds the collection of high dimensional data, and is sent to data handling system 2 the new high dimensional data collected;

Step 7: after data handling system 2 receives the new high dimensional data collected, is then stored in high dimensional data for depositing in the queue 4 of high dimensional data according to the sequencing received successively;

Step 8: then data handling system 2 takes out a new high dimensional data collected from the queue 4 for depositing high dimensional data sequentially successively, takes out a new high dimensional data X collected _newafter, just carry out this high dimensional data X _newthe extraction of characteristic component and dimensionality reduction operation, described high dimensional data X _newbe expressed as (x ' ₁, x ' ₂... x ' _n), n is the dimension of this new high dimensional data;

Step 9: described carrying out operates to comprise to the extraction of the characteristic component of this new high dimensional data collected and dimensionality reduction and be first converted to the module 3 of low-dimensional data this new high dimensional data X collected for the high dimensional data of increment type _newproject to b ₁, b ₂b _kfor in the k dimension space of substrate, the high dimensional data X that what this was new collect _newproject to b ₁, b ₂b _kfor the method in the k dimension space of substrate obtains result vector r according to formula (3) with alternative manner _k:

$r_k=X_{\mathrm{new}}-\underset{i=1}{\overset{k}{\mathrm{\Σ}}}{b}_i{b}_i^T{X}_{\mathrm{new}}---\left(3\right)$

The span of described i is be the dimension of current spatial object S from 1 to k, k, is also the dimension of low-dimensional after current Data Dimensionality Reduction, r _kfor result vector, when the length of result vector || r _k|| ₂be less than T _ktime, k value is constant, and current spatial object S also remains unchanged, || r _k|| ₂be more than or equal to T _ktime, try to achieve kth+1 coefficient by b _k+1add former k dimension space object S as new spatial base, make the dimension of spatial object S increase by 1, k=k+1.Setting r ₀=X _new, and r _ifor intermediate vector, T _ibe the i-th threshold value, x _maxfor that high dimensional data that data length in the queue 4 of the high dimensional data of current system process is the longest;

Step 10: according to obtained b ₁, b ₂b _kobtained for this high dimensional data X by formula (4) _newdimensionality reduction data vector in the part Y of component non-zero _new:

$Y_{\mathrm{new}}=\left[\begin{array}{c}{X_{\mathrm{new}}}^T{b}_1\\ {X_{\mathrm{new}}}^T{b}_2\\ ...\\ {X_{\mathrm{new}}}^T{b}_i\\ ....\\ {X_{\mathrm{new}}}^T{b}_k\end{array}\right]---\left(4\right)$

After the queue of high dimensional data is all disposed, according to the value of final k, by the dimensionality reduction data vector rear portion of whole processed high dimensional data add component be the part of zero, be collectively expressed as k dimension dimensionality reduction after data vector.

Apply such scheme of the present invention, can also reach the redundance eliminating the such high dimensional data of video, audio frequency, weather and view data that data acquisition system is come, the complexity simplifying high dimensional data, the immanent structure disclosing high dimensional data and contact, raising dimension data treatment effeciency, improve dimensionality reduction after data intelligibility and after improving dimensionality reduction data accurately reflect the effect of original high dimensional data.

Accompanying drawing explanation

Figure l is theory structure schematic diagram of the present invention.

Fig. 2 is method of the present invention for the high dimensional data of first group of increment type with the design sketch of the contrast reconstructed error of two kinds of methods of prior art.

Fig. 3 is method of the present invention for the high dimensional data of second group of increment type with the design sketch of the contrast reconstructed error of two kinds of methods of prior art.

Fig. 4 is method of the present invention for the high dimensional data of the 3rd group of increment type with the design sketch of the contrast reconstructed error of two kinds of methods of prior art.

Fig. 5 is method of the present invention for the high dimensional data of the 4th group of increment type with the design sketch of the contrast reconstructed error of two kinds of methods of prior art.

Fig. 6 is method of the present invention for the high dimensional data of the 5th group of increment type with the design sketch of the contrast reconstructed error of two kinds of methods of prior art.

Fig. 7 is method of the present invention for the high dimensional data of the 6th group of increment type with the design sketch of the contrast reconstructed error of two kinds of methods of prior art.

Fig. 8 is method of the present invention for the high dimensional data of first group of increment type with the contrast dimensionality reduction of two kinds of methods of prior art design sketch consuming time.

Fig. 9 is method of the present invention for the high dimensional data of second group of increment type with the contrast dimensionality reduction of two kinds of methods of prior art design sketch consuming time.

Figure 10 is method of the present invention for the high dimensional data of the 3rd group of increment type with the contrast dimensionality reduction of two kinds of methods of prior art design sketch consuming time.

Figure 11 is method of the present invention for the high dimensional data of the 4th group of increment type with the contrast dimensionality reduction of two kinds of methods of prior art design sketch consuming time.

Figure 12 is method of the present invention for the high dimensional data of the 5th group of increment type with the contrast dimensionality reduction of two kinds of methods of prior art design sketch consuming time.

Figure 13 is method of the present invention for the high dimensional data of the 6th group of increment type with the contrast dimensionality reduction of two kinds of methods of prior art design sketch consuming time.

Embodiment

Most existing dimension reduction method all needs user to set the dimension (target dimension) of feature space, a lot of dimension reduction method does not possess the ability that online increment is extended, the method step so just repeating dimensionality reduction is many, cause the resource of the system that takies also many, and the time complexity increased, many traditional dimensions about subtract method, and needing derives proper vector or carry out matrix inversion operation, need larger time complexity or cause the instability of algorithm.

Below in conjunction with accompanying drawing, summary of the invention is described further:

With reference to shown in Fig. 1, the high dimensional data of increment type is converted to the system of low-dimensional data, comprise high dimensional data acquisition system 1, described high dimensional data acquisition system 1 is connected with data handling system 2, the high dimensional data included in described data handling system 2 for increment type is converted to the module 3 of low-dimensional data, includes the queue 4 for depositing high dimensional data in described data handling system 2.

The high dimensional data of described increment type is converted to the method for the system of low-dimensional data, as follows:

Step 1: first high dimensional data acquisition system carries out the collection for the such high dimensional data of video, audio frequency, weather or view data, is then sent to data handling system 2 by the high dimensional data collected;

Step 2: after data handling system 2 receives high dimensional data, then according to the sequencing received, high dimensional data is stored in for depositing in the queue 4 of high dimensional data successively, start and be used for the module 3 that the high dimensional data of increment type is converted to low-dimensional data and set a n-dimensional space object V, include k dimension space object S in described n-dimensional space object V, k is initially set to 0;

Step 3: then data handling system 2 takes out a high dimensional data from the queue 4 for depositing high dimensional data sequentially successively, after taking out a high dimensional data X, just carry out the extraction to the characteristic component of this high dimensional data X and dimensionality reduction operation, described high dimensional data X is expressed as (x ₁, x ₂... x _n), n is the dimension of this high dimensional data;

Step 4: described carrying out comprises to the extraction of the characteristic component of this high dimensional data and dimensionality reduction operation the module 3 being first converted to low-dimensional data for the high dimensional data of increment type and this high dimensional data X is projected in the k dimension space represented by k dimension space object S, and this high dimensional data X method projected in the k dimension space S represented by k dimension space object obtains result vector r according to formula (1) with alternative manner _k:

$r_k=X-\underset{i=1}{\overset{k}{\mathrm{\Σ}}}{b}_i{b}_i^{T}X---\left(1\right)$

The span of described i is be the dimension of current spatial object S from 1 to K, K, is also the dimension of low-dimensional after current Data Dimensionality Reduction, the first coefficient r _kfor result vector, when the length of result vector || r _k|| ₂be less than T _ktime, k value is constant, and current spatial object S also remains unchanged, || r _k|| ₂be more than or equal to T _ktime, try to achieve kth+1 coefficient by b _k+1add former k dimension space object S as new spatial base, make the dimension of spatial object S increase by 1, k=k+1.Setting r ₀=X, and r _ifor intermediate vector, T _ibe the i-th threshold value, x _maxfor that high dimensional data that data length in the queue 4 for depositing high dimensional data is the longest;

Step 5: according to obtained b ₁, b ₂b _kthe part Y of component non-zero in the dimensionality reduction data vector obtaining for this high dimensional data X by formula (2):

$Y=\left[\begin{array}{c}{X}^T{b}_1\\ X^T{b}_2\\ ...\\ X^T{b}_i\\ ....\\ X^T{b}_k\end{array}\right]---\left(2\right)$

After the queue of high dimensional data is all disposed, according to the value of final k, the part of the dimensionality reduction data vector component non-zero of whole high dimensional data is supplemented the part that upper component is zero, be collectively expressed as k dimension dimensionality reduction after data vector.

Step 6: after obtaining the dimensionality reduction data vector for each high dimensional data, if high dimensional data acquisition system proceeds the collection of high dimensional data, and is sent to data handling system 2 the new high dimensional data collected;

Step 7: after data handling system 2 receives the new high dimensional data collected, is then stored in high dimensional data for depositing in the queue 4 of high dimensional data according to the sequencing received successively;

Step 8: then data handling system 2 takes out a new high dimensional data collected from the queue 4 for depositing high dimensional data sequentially successively, takes out a new high dimensional data X collected _newafter, just carry out this high dimensional data X _newthe extraction of characteristic component and dimensionality reduction operation, described high dimensional data X _newbe expressed as (x ' ₁, x ' ₂... x ' _n), n is the dimension of this new high dimensional data;

Step 9: described carrying out operates to comprise to the extraction of the characteristic component of this new high dimensional data collected and dimensionality reduction and be first converted to the module 3 of low-dimensional data this new high dimensional data X collected for the high dimensional data of increment type _newproject to b ₁, b ₂b _kfor in the k dimension space of substrate, the high dimensional data X that what this was new collect _newproject to b ₁, b ₂b _kfor the method in the k dimension space of substrate obtains result vector r according to formula (3) with alternative manner _k:

$r_k=X_{\mathrm{new}}-\underset{i=1}{\overset{k}{\mathrm{\Σ}}}{b}_i{b}_i^T{X}_{\mathrm{new}}---\left(3\right)$

The span of described i is be the dimension of current spatial object S from 1 to k, k, is also the dimension of low-dimensional after current Data Dimensionality Reduction, r _kfor result vector, when the length of result vector || r _k|| ₂be less than T _ktime, k value is constant, and current spatial object S also remains unchanged, || r _k|| ₂be more than or equal to T _ktime, try to achieve kth+1 coefficient by b _k+1add former k dimension space object S as new spatial base, make the dimension of spatial object S increase by 1, k=k+1.Setting r ₀=X _new, and r _ifor intermediate vector, T _ibe the i-th threshold value, x _maxfor that high dimensional data that data length in the queue 4 of the high dimensional data of current system process is the longest;

Step 10: according to obtained b ₁, b ₂b _kobtained for this high dimensional data X by formula (4) _newdimensionality reduction data vector in the part Y of component non-zero _new:

$Y_{\mathrm{new}}=\left[\begin{array}{c}{X_{\mathrm{new}}}^T{b}_1\\ {X_{\mathrm{new}}}^T{b}_2\\ ...\\ {X_{\mathrm{new}}}^T{b}_i\\ ....\\ {X_{\mathrm{new}}}^T{b}_k\end{array}\right]---\left(4\right)$

After the queue of high dimensional data is all disposed, according to the value of final k, by the dimensionality reduction data vector rear portion of whole processed high dimensional data add component be the part of zero, be collectively expressed as k dimension dimensionality reduction after data vector.

Method of the present invention can solve the shortcoming of most existing dimension reduction method, thus realize determining that target dimension, online increment are extended and without the need to deriving proper vector secular equation or carrying out matrix inversion operation, obtain orthogonal component vector with the quantity of as far as possible little derived data adaptively.And IOCA computation complexity is O (Ndk), N is data amount check, and d is raw data dimension, k is that target dimension IOCA only need travel through a secondary data, and the low-dimensional that just simultaneously can obtain orthogonal component and data represents, if b ₁, b ₂..., b _kfor the orthogonal basement finally obtained, the present invention can ensure like this for each high dimensional data, the result data after all making dimensionality reduction is less than with the error of raw data after reconstruct as Fig. 2, Fig. 3, Fig. 4, Fig. 5, shown in Fig. 6 and Fig. 7, these six accompanying drawings respectively show the design sketch of dimension reduction method reconstructed error after the dimensionality reduction of the high dimensional data for six groups of increment types with the IPCA dimension reduction method in existing technology and CCIPCA dimension reduction method of the IOCA representated by the present invention, as can be seen from the figure the dimension error after the dimensionality reduction automatically determined of method of the present invention is little, dimension after the method dimensionality reduction of other prior art is uncertain, the accurate reproduction of data after dimensionality reduction cannot be ensured, and the method for prior art cannot ensure the orthogonality of basis vector as method of the present invention.

As shown in Fig. 8, Fig. 9, Figure 10, Figure 11, Figure 12 and Figure 13, the dimension reduction method that these six accompanying drawings respectively show the IOCA representated by the present invention is with the IPCA dimension reduction method in existing technology and the CCIPCA dimension reduction method design sketch consuming time at the dimensionality reduction of the high dimensional data for six groups of increment types, and as can be seen from the figure method of the present invention carries out the two kinds of methods being far smaller than other prior aries consuming time of increment type dimensionality reduction computing.

The above, it is only preferred embodiment of the present invention, not any pro forma restriction is done to the present invention, although the present invention discloses as above with preferred embodiment, but and be not used to limit the present invention, any those skilled in the art, do not departing within the scope of technical solution of the present invention, make a little change when the technology contents of above-mentioned announcement can be utilized or be modified to the Equivalent embodiments of equivalent variations, in every case be do not depart from technical solution of the present invention content, according to technical spirit of the present invention, within the spirit and principles in the present invention, to any simple amendment that above embodiment is done, equivalent replacement and improvement etc., within the protection domain all still belonging to technical solution of the present invention.

Claims (2)

1. the high dimensional data of an increment type is converted to the system of low-dimensional data, it is characterized in that comprising high dimensional data acquisition system, described high dimensional data acquisition system is connected with data handling system, the high dimensional data included in described data handling system for increment type is converted to the module of low-dimensional data, includes the queue for depositing high dimensional data in described data handling system.

2. the high dimensional data of increment type according to claim 1 is converted to the method for the system of low-dimensional data, it is characterized in that, as follows:

Step 1: first high dimensional data acquisition system carries out the collection for the such high dimensional data of video, audio frequency, weather or view data, is then sent to data handling system by the high dimensional data collected;

Step 2: after data handling system receives high dimensional data, then according to the sequencing received, high dimensional data is stored in for depositing in the queue of high dimensional data successively, start and be used for the module that the high dimensional data of increment type is converted to low-dimensional data and set a n-dimensional space object V, include k dimension space object S in described n-dimensional space object V, k is initially set to 0;

Step 3: then data handling system takes out a high dimensional data from the queue for depositing high dimensional data sequentially successively, after taking out a high dimensional data X, just carry out the extraction to the characteristic component of this high dimensional data X and dimensionality reduction operation, described high dimensional data X is expressed as (x ₁, x ₂... x _n), n is the dimension of this high dimensional data;

Step 4: described carrying out comprises to the extraction of the characteristic component of this high dimensional data and dimensionality reduction operation the module being first converted to low-dimensional data for the high dimensional data of increment type and this high dimensional data X is projected in the k dimension space represented by k dimension space object S, and this high dimensional data X method projected in the k dimension space S represented by k dimension space object obtains result vector r according to formula (1) with alternative manner _k:

$r_k=X-\underset{i=1}{\overset{k}{\mathrm{\Σ}}}{b}_i{b}_i^{T}X---\left(1\right)$

The span of described i is be the dimension of current spatial object S from 1 to k, k, is also the dimension of low-dimensional after current Data Dimensionality Reduction, the first coefficient r _kfor result vector, when the length of result vector || r _k|| ₂be less than T _ktime, k value is constant, and current spatial object S also remains unchanged, || r _k|| ₂be more than or equal to T _ktime, try to achieve kth+1 coefficient by b _k+1add former k dimension space object S as new spatial base, make the dimension of spatial object S increase by 1, k=k+1.Setting r ₀=X, and r _ifor intermediate vector, T _ibe the i-th threshold value, x _maxfor that high dimensional data that data length in the queue 4 for depositing high dimensional data is the longest;

Step 5: according to obtained b ₁, b ₂b _kthe part Y of component non-zero in the dimensionality reduction data vector obtaining for this high dimensional data X by formula (2):

$Y=\left[\begin{array}{c}{X}^T{b}_1\\ X^T{b}_2\\ ...\\ X^T{b}_i\\ ....\\ X^T{b}_k\end{array}\right]---\left(2\right)$

After the queue of high dimensional data is all disposed, according to the value of final k, the part of the dimensionality reduction data vector component non-zero of whole high dimensional data is supplemented the part that upper component is zero, be collectively expressed as k dimension dimensionality reduction after data vector.

Step 6: after obtaining the dimensionality reduction data vector for each high dimensional data, if high dimensional data acquisition system proceeds the collection of high dimensional data, and is sent to data handling system 2 the new high dimensional data collected;

Step 7: after data handling system receives the new high dimensional data collected, is then stored in high dimensional data for depositing in the queue of high dimensional data according to the sequencing received successively;

Step 8: then data handling system takes out a new high dimensional data collected from the queue for depositing high dimensional data sequentially successively, takes out a new high dimensional data X collected _newafter, just carry out this high dimensional data X _newthe extraction of characteristic component and dimensionality reduction operation, described high dimensional data X _newbe expressed as (x ' ₁, x ' ₂... x ' _n), n is the dimension of this new high dimensional data;

Step 9: described carrying out operates to comprise to the extraction of the characteristic component of this new high dimensional data collected and dimensionality reduction and be first converted to the module of low-dimensional data this new high dimensional data X collected for the high dimensional data of increment type _newproject to b ₁, b ₂b _kfor in the k dimension space of substrate, the high dimensional data X that what this was new collect _newproject to b ₁, b ₂b _kfor the method in the k dimension space of substrate obtains result vector r according to formula (3) with alternative manner _k:

$r_k=X_{\mathrm{new}}-\underset{i=1}{\overset{k}{\mathrm{\Σ}}}{b}_i{b}_i^T{X}_{\mathrm{new}}---\left(3\right)$

The span of described i is be the dimension of current spatial object S from 1 to k, k, is also the dimension of low-dimensional after current Data Dimensionality Reduction, r _kfor result vector, when the length of result vector || r _k|| ₂be less than T _ktime, k value is constant, and current spatial object S also remains unchanged, || r _k|| ₂be more than or equal to T _ktime, try to achieve kth+1 coefficient by b _k+1add former k dimension space object S as new spatial base, make the dimension of spatial object S increase by 1, k=k+1.Setting r ₀=X _new, and r _ifor intermediate vector, T _ibe the i-th threshold value, x _maxfor that high dimensional data that data length in the queue 4 of the high dimensional data of current system process is the longest;

Step 10: according to obtained b ₁, b ₂b _kobtained for this high dimensional data X by formula (4) _newdimensionality reduction data vector in the part Y of component non-zero _new:

$Y_{\mathrm{new}}=\left[\begin{array}{c}{X_{\mathrm{new}}}^T{b}_1\\ {X_{\mathrm{new}}}^T{b}_2\\ ...\\ {X_{\mathrm{new}}}^T{b}_i\\ ....\\ {X_{\mathrm{new}}}^T{b}_k\end{array}\right]---\left(4\right)$

After the queue of high dimensional data is all disposed, according to the value of final k, by the dimensionality reduction data vector rear portion of whole processed high dimensional data add component be the part of zero, be collectively expressed as k dimension dimensionality reduction after data vector.