CN110058841B - General computing device and method for nonlinear function with symmetry - Google Patents
General computing device and method for nonlinear function with symmetry Download PDFInfo
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Abstract
The invention provides a general computing device and a method for a nonlinear function with symmetry. The calculating device comprises an absolute value taking unit, an address index unit, a lookup table unit, a multiplication and addition unit, a symbol taking unit, a subtracter and a selector. The address index unit comprises a comparator, a controller, an address register and a segment endpoint memory. The device for calculating the nonlinear function with symmetry can calculate any nonlinear function value with axial symmetry or central symmetry, and has certain universality. Meanwhile, the method carries out segmentation processing on the original nonlinear function according to the maximum absolute error of the target piecewise linear function and the original nonlinear function, and can realize controllable precision of an approximate calculation result.
Description
Technical Field
The invention relates to a general computing device and method for a nonlinear function with symmetry, belonging to the technical field of digital signal processing.
Background
The nonlinear function is a more complex function, and the general methods for processing the nonlinear function include taylor expansion, CORDIC algorithm, etc. Although these algorithms can obtain a high-accuracy operation result when performing operation of a nonlinear function, the calculation is complicated and the operation is complicated.
In recent years, with the development of artificial intelligence technology, various deep neural network models have been proposed by researchers. However, the operation of nonlinear functions, such as sigmoid function and tanh function, is often involved in the neural network model, and the operation of these nonlinear functions often limits the operation speed and overall performance of the whole neural network. On the other hand, the neural network algorithm has no high requirement on the accuracy of the operation result of the nonlinear functions, allows the nonlinear functions to be approximately calculated, and the nonlinear functions such as sigmoid and tanh are often symmetrical, and can fully utilize the inherent symmetry of the nonlinear functions to quickly calculate.
Disclosure of Invention
In order to obtain a faster operation speed and use less circuit resources, the invention provides a general computing device and method for a nonlinear function with symmetry in a neural network algorithm.
The technical scheme of the device is as follows:
the general-purpose computing device for the nonlinear function with symmetry comprises: an absolute value taking unit, configured to perform an absolute value taking operation on input data; the address indexing unit is used for determining a linear interval in which the absolute value of the input data is positioned according to the absolute value of the input data and an interval endpoint value of the linear segmentation of the original nonlinear function; the lookup table unit is used for finding out the parameter values of the linear function according to the address index values of the linear intervals in which the absolute values of the input data are positioned, wherein the parameter values comprise a slope k and a y-axis intercept h; the multiplication and addition unit is used for calculating the output of the linear function corresponding to the absolute value of the input data according to the absolute value of the input data and the slope k and intercept h of the linear function corresponding to the absolute value of the input data; a sign bit unit for determining the sign of the original input data; the subtracter is used for carrying out subtraction operation on the bias constant of the coordinate value y of the central point of the nonlinear function and the output result of the multiplication and addition unit to obtain a corresponding output value when the input data is a negative number; a selector for determining a final output value based on a symmetric pattern of the non-linear function and a sign bit of the input data.
The invention relates to a general calculation method for a nonlinear function with symmetry, which comprises the following steps: (1) Configuring a corresponding symmetric mode according to the symmetric type of the nonlinear function, wherein the symmetric type comprises central symmetry and axial symmetry; (2) If the non-linear function is of the type of central symmetry, the point of central symmetry is determined to be (0,y) bias ),And configuring the bias constant of the y coordinate of the central point to be 2 x y bias Wherein y is bias A y coordinate representing a symmetric center point of the nonlinear function; if the nonlinear function is of an axisymmetric type, configuring a bias constant of a y coordinate of the central point to be 0; (3) For x in the original non-linear function f (x)>The part 0 is subjected to linear segmentation processing, and the end point value of the obtained linear segmentation interval and the linear function parameter value corresponding to the linear segmentation interval are stored, wherein the parameter value comprises a slope k and a y-axis intercept h; (4) Taking an absolute value of input data of the nonlinear function, and judging a sign of the input data; (5) Comparing the absolute value of the input data with the endpoint value of the linear segmentation interval, and outputting the index number of the linear segmentation interval; (6) Obtaining the slope k and intercept h of the corresponding piecewise linear function according to the index number in the step (5) by using a table look-up method; (7) Carrying out multiplication and addition operation on the absolute value of the input data, the slope k of the piecewise linear function and the intercept h; (8) And (4) determining a final output result according to the symmetrical mode configured in the step (1) and the symbol of the input data determined in the step (4).
The invention has the following beneficial effects:
(1) The device for calculating the nonlinear function with symmetry can calculate any nonlinear function value with axial symmetry or central symmetry, and has certain universality.
(2) The approximate calculation is carried out by carrying out piecewise linearity on the original nonlinear function, and the approximate calculation is combined with the lookup table, so that the calculation speed is improved, and the consumption of calculation resources is reduced.
(3) The symmetry of the original nonlinear function is fully utilized, only half of the storage space of the segmented endpoint memory of the address index unit and the storage space of the lookup table are needed, and the consumption of storage resources is greatly reduced.
(4) And the original nonlinear function is processed in a segmented manner according to the maximum absolute error of the target piecewise linear function and the original nonlinear function, so that the accuracy of the approximate calculation result can be controlled.
Drawings
Fig. 1 is an overall architecture diagram of a nonlinear function calculation apparatus oriented to have symmetry.
Fig. 2 is a schematic diagram of the internal structure of the address indexing unit.
Fig. 3 is a control flow diagram of the address indexing unit.
Fig. 4 is a schematic diagram of the internal structure of the multiply-add unit.
Fig. 5 is a flow chart for linear segmentation of a nonlinear function in terms of maximum absolute error.
Detailed Description
The following describes the present invention in detail with reference to the accompanying drawings.
FIG. 1 is a schematic structural diagram of a computing device with a symmetric nonlinear function according to the present invention, the device including: an absolute value taking unit, configured to take an absolute value operation on an input value x; the address indexing unit is used for determining a linear interval where an input absolute value is located according to the input absolute value and an endpoint value of a piecewise linear interval of an original nonlinear function; the lookup table unit is used for finding out the parameter values of the linear function corresponding to the input absolute value according to the address index of the linear interval, wherein the parameter values comprise a slope k and a y-axis intercept h; the multiplication and addition unit is used for calculating the output of a linear function corresponding to the input absolute value according to the input absolute value and the linear function parameter; a sign bit taking unit for judging the sign of the original input value x; the subtracter is used for carrying out subtraction operation on the offset constant of the coordinate value of the central point y and the output result of the multiplication and addition unit to obtain a corresponding output value when the input value x is a negative number; a selector for determining a final output value depending on whether the symmetry mode is axisymmetric or centrosymmetric and the sign bit of the input value x.
The address indexing unit includes: the comparator is used for comparing the input absolute value with the linear segmented interval endpoint value in size; the controller is used for controlling the address register to carry out accumulation operation or output a final address index value and reset according to the result obtained by the comparator; the address register is used for receiving an accumulation or reset command of the controller to carry out accumulation or reset operation; and the segmented endpoint memory is used for taking out the endpoint value of the corresponding linear function interval according to the address register.
The multiplication and addition unit includes: the multiplier is used for multiplying the input absolute value and a slope parameter k of the linear function; and the adder is used for adding the multiplier result and the vertical coordinate intercept h of the linear function.
According to another aspect of the present invention, there is provided a method for calculating a nonlinear function with symmetry, comprising the steps of:
(1) Judging whether the nonlinear function is centrosymmetric or axisymmetric, and configuring a corresponding symmetric mode;
(2) If the non-linear function is centrosymmetric, then determine the centrosymmetric point (0, y) bias ) And configuring the bias constant of the y coordinate of the central point to be 2 x y bias If not, the bias constant is set to 0;
(3) Performing piecewise linear processing on a part where x is greater than 0 in an original nonlinear function f (x), storing an obtained linear piecewise interval endpoint value in a piecewise endpoint memory of an address index unit, and storing linear function parameters k and h corresponding to a linear interval in a lookup table unit;
(4) Performing multiply-add calculation by using the input absolute value of the nonlinear function and the parameters k and h of the corresponding linear function;
(5) And determining a final output result according to the symmetrical mode and the sign bit of the nonlinear function input.
This embodiment describes the linear segmentation method of the nonlinear function with symmetry as follows:
if a function f (x) is given by (0, y) bias ) With central symmetry, it will have the following properties:
furthermore, if a function f (x) is axisymmetric with the y-axis, its properties are as follows:
that is, whether f (x) is centrosymmetric or axialSymmetry (if the center of symmetry is not on the y-axis or the axis of symmetry is not the y-axis f (x) can be made (0,y) by biasing the x variable bias ) Centrosymmetric or axisymmetric with the y-axis), only the value f (| x |) of the positive half axis needs to be calculated, and the value of the negative half axis has correlation with f (| x |). Therefore, when the nonlinear function f (x) with symmetry is subjected to piecewise linear processing, only the positive half-axis image of f (x) needs to be subjected to piecewise processing, which saves a large amount of storage space and calculation time.
On the other hand, when the nonlinear function is subjected to piecewise linear processing, the accuracy controllability of a final approximate result is fully considered, and the maximum absolute error between the piecewise linear function value and the original nonlinear function value is ensured to be smaller than a set error value.
The effective interval of the nonlinear function f (x) is assumed to be [ x d o wn ,x up ]The maximum absolute error of the approximation result is set to max err o r And the iteration number of the segmentation is a variable i, the specific segmentation process is as follows (see fig. 5):
first, setting initial interval lower limit and interval upper limit and initial value of i, x 0 =x d o wn ,x n =x up ,i=0;
Second step, connect (x) 0 ,f(x 0 ) And (x) n ,f(x n ) These two points) form a linear function f _ line (x), and the linear function and the original nonlinear function are calculated at [ x ] 0 ,x n ]The maximum absolute error max (| f (x) -f _ line (x) |) in the interval is compared with the set max error If it is larger than the set maximum absolute error max error The third step is changed, otherwise, the fourth step is changed;
third, reducing the interval upper limit x n A value of (a), let x n =x n –min step (min step Is the upper limit x of the interval n Decreasing minimum step size), go to the operation of the second step;
the fourth step, judge x at this moment n Whether or not to be equal to x up If so, the iterative process ends immediately, otherwise, the process turns toThe fifth step;
fifthly, recording x at the moment n End point value of (1), let x i =x n (ii) a And recording the parameter information, slope k, of the corresponding linear function f _ line (x) i And y-axis intercept h i ;
Sixthly, changing the lower limit and the upper limit of the interval, x 0 =x i ,x n =x up And adding 1 to the iteration number, enabling i = i +1, and turning to the operation of the second step.
Taking sigmoid (x) = 1./(1 + exp (-x)) as an example, the valid interval is [ -10, 10 [ -10 +]The maximum absolute error of the approximation is set to 0.01, the minimum step min step Set to 0.001, since the sigmoid function is centrosymmetric with (0, 0.5), the effective interval can be reduced to [0, 10 ]]After the above segmentation processing, the obtained endpoint values and the parameters of the corresponding linear functions are shown in table 1:
TABLE 1 Interval and corresponding Linear function parameters in the iterative Process
Number of iterations i | Endpoint value x i | Segment interval | Slope parameter (k) i ) | y-axis intercept parameter (h) i ) |
0 | 1.1360 | [0,1.1360] | 0.2262 | 0.5000 |
1 | 2.0700 | [1.1360,2.0700] | 0.1403 | 0.5976 |
2 | 3.3000 | [2.0700,3.3000] | 0.0622 | 0.7592 |
3 | 6.0090 | [3.3000,6.0090] | 0.0122 | 0.9241 |
4 | 10 | [6.0090,10] | 0.0006 | 0.9939 |
/ | ∞ | [10,∞] | 0 | 1 |
Examples
Assuming that the calculation of the value of the nonlinear function sigmoid at x = -7 by the calculating device of the present invention, the following processing units are sequentially performed:
absolute value taking unit: the absolute value is taken for the input value. In the embodiment of the present invention, if x = -7 is input, the output is | x | =7.
An address indexing unit: as shown in fig. 2, includes a comparator, a controller, an address register, and a segment endpoint memory. Firstly, initializing assignment is required to be carried out on a segmentation endpoint memory, and in the embodiment of the invention, 10,6.0090,3.3000,2.0700 and 1.1360 of endpoint values of the sigmoid function segmentation are sequentially stored in the memory from large to small; then, corresponding operations are performed according to the control flow chart shown in fig. 3: comparing the output 7 of the absolute value unit with the initial endpoint 10, if the comparison result is less than 10, adding 1 to the address register (the initial value is 0), taking the next endpoint value of 6.0090, and comparing again, if the output 7 of the absolute value unit is greater than the endpoint value of 6.0090, outputting the address index value at the moment, if the result is 1, and resetting the address register.
A lookup table unit: firstly, linear function parameters corresponding to the segmented linear function need to be stored in a lookup table, in this embodiment of the present invention, values (ki, hi) of (0, 1), (0.0006, 0.9939), (0.0122, 0.9241), (0.0622, 0.7592), (0.1403, 0.5976), and (0.2262, 0.5000), respectively, are sequentially stored in the lookup table, and then parameters of the nonlinear function are determined according to an output value of the address indexing unit, and since an output address index value of the address indexing unit is 1, parameters of the searched linear function are (0.0006, 0.9939), that is, k =0.0006, h =0.9939.
A multiplication and addition unit: as shown in fig. 4, the value of the k x + h linear result is calculated. In the embodiment of the present invention, the corresponding linear function output is calculated according to the input absolute value | x | =7 and the parameter k =0.0006, h =0.9939 of the linear function, and the result is 0.0006 | + 7+0.9939=0.9981.
A symbol taking unit: if the input is a positive number, the output sign is 0, otherwise, the output is 1. In an embodiment of the present invention, the input is-7, which is a negative number, and the output is 1.
A subtracter: the value of the multiply-add unit is subtracted from the center point y coordinate offset constant. First, it is necessary to determine the offset constant of the y coordinate of the center point, in the embodiment of the present invention, since the sigmoid function is centered at (0, 0.5), the offset constant of the y coordinate of the center point is 2 × 0.5=1, and then the result of the multiplication and addition unit is subtracted from the offset constant, so the result of the subtractor is 1-0.9981=0.0019.
A selector: and determining final result output according to the symmetrical mode and the input sign bit. Firstly, a symmetric mode is configured, and if the symmetric mode is axisymmetric, the mode =0; mode =1 if centrosymmetric. In the embodiment of the invention, the sigmoid function is centrosymmetric, so that the symmetric mode is 1. Then, the final output value is jointly determined according to the input sign and the symmetric model mode:
in the embodiment of the present invention, mode =1 and sign =1, the final approximation result is 0.0019 of the subtractor, and the value of the original sigmoid function at x = -7 is 0.0009, and the final result error is 0.001, which is smaller than the initially set maximum error value of 0.01, so that the operation of the nonlinear function with controllable precision is realized.
Claims (4)
1. The general-purpose computing device for nonlinear functions with symmetry is characterized by comprising:
the absolute value taking unit is used for carrying out absolute value taking operation on input data;
the address indexing unit is used for determining a linear interval in which the absolute value of the input data is positioned according to the absolute value of the input data and an interval endpoint value of the linear segmentation of the original nonlinear function;
the lookup table unit is used for finding out parameter values of the linear function according to the address index value of the linear interval where the absolute value of the input data is located, wherein the parameter values comprise a slope k and a y-axis intercept h;
the multiplication and addition unit is used for calculating the output of the linear function corresponding to the absolute value of the input data according to the absolute value of the input data and the slope k and intercept h of the linear function corresponding to the absolute value of the input data;
a sign bit unit for determining the sign of the original input data;
the subtractor is used for carrying out subtraction operation on the offset constant of the coordinate value y of the central point of the nonlinear function and the output result of the multiplication and addition unit to obtain a corresponding output value when the input data is a negative number;
a selector for determining a final output value based on a symmetry mode of the non-linear function and a sign bit of the input data;
the address indexing unit includes:
the comparator is used for comparing the absolute value of the input data with the linear piecewise interval endpoint value of the original nonlinear function in size;
the controller is used for controlling the relevant operation of the address register according to the result obtained by the comparator, outputting the value of the address register at the moment if the absolute value of the input data is greater than the endpoint value, and resetting the address register; if the absolute value of the input data is less than or equal to the endpoint value, performing accumulation operation on the address register;
the address register is used for receiving an accumulation or reset command of the controller to perform accumulation or reset operation;
and the segmented endpoint memory is used for taking out the endpoint value of the corresponding linear function interval according to the address register.
2. The general calculation method for the nonlinear function with symmetry is characterized by comprising the following steps of:
(1) Configuring a corresponding symmetric mode according to the symmetric type of the nonlinear function, wherein the symmetric type comprises central symmetry and axial symmetry;
(2) If the non-linear function is of the type of central symmetry, the point of central symmetry is determined to be (0,y) bias ) And configuring the bias constant of the y coordinate of the central point to be 2 x y bias Wherein y is bias A y coordinate representing a symmetric center point of the nonlinear function; if the nonlinear function is of an axisymmetric type, configuring a bias constant of a central point y coordinate to be 0;
(3) Performing linear segmentation processing on a part x >0 in an original nonlinear function f (x), and storing an end point value of an obtained linear segmentation interval and a linear function parameter value corresponding to the linear segmentation interval, wherein the parameter value comprises a slope k and a y-axis intercept h; the linear segmentation processing is to segment according to the maximum absolute error of the final piecewise linear function and the original nonlinear function;
(4) Taking absolute values of input data of the nonlinear function, and judging signs of the input data;
(5) Comparing the absolute value of the input data with the endpoint value of the linear segmentation interval, and outputting the index number of the linear segmentation interval;
(6) Obtaining the slope k and intercept h of the corresponding piecewise linear function according to the index number in the step (5) by using a table look-up method;
(7) Carrying out multiplication and addition operation on the absolute value of the input data, the slope k of the piecewise linear function and the intercept h;
(8) And (4) determining a final output result according to the symmetrical mode configured in the step (1) and the symbol of the input data determined in the step (4).
3. The method for calculating nonlinear function with symmetry as claimed in claim 2, wherein the effective interval of the nonlinear function f (x) is assumed to be [ x ] in linear piecewise processing down ,x up ]The maximum absolute error of the approximation result is set to max error And the iteration number of the segment is a variable i, the specific processing procedure is as follows:
first, setting an interval lower limit variable x 0 Interval upper limit variable x n And the initial value of i, x 0 =x down ,x n =x up ,i=0;
Second step, connect (x) 0 ,f(x 0 ) And (x) n ,f(x n ) These two points) form a linear function f _ line (x), and the linear function and the original nonlinear function f (x) are calculated in [ x ] 0 ,x n ]The maximum absolute error max (| f (x) -f _ line (x) |) in the interval is compared with the set maximum absolute error max error If it is larger than the set maximum absolute error max error Then transfer to the third stepOtherwise, turning to the fourth step;
thirdly, reducing the interval upper limit variable x n A value of (a), let x n =x n –min step ,min step Is the upper limit x of the interval n Decreasing the minimum step size, and then turning to the second step;
the fourth step, judge the interval upper limit variable x at this moment n Whether or not to be equal to x up If the values are equal, the iteration process is immediately finished, otherwise, the step is changed to the fifth step;
fifthly, recording the interval upper limit variable x at the moment n End point value of (c), let x i =x n (ii) a And recording the parameter information of the corresponding linear function f _ line (x), the slope k i And y-axis intercept h i ;
Sixthly, changing the interval lower limit variable x 0 And an interval upper limit variable x n A value of (a) x 0 =x i And x n =x up The number of iterations is increased by 1, let i = i +1, and then go to the second step.
4. The general calculation method for nonlinear functions with symmetry as claimed in claim 2, wherein in step (5), if the absolute value of the input data is greater than the endpoint value, the index number of the linear segment interval at that time is output; and (5) if the absolute value of the input data is less than or equal to the endpoint value, adding 1 to the index number, comparing the absolute value of the input data with the size of the endpoint value of the next segmentation interval, and repeating the operation in the step (5).
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