| Name | chiku JSON |
| Version |
0.0.25
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| Summary | Python package for efficient probabilistic polynomial approximation of arbitrary functions. |
| upload_time | 2023-07-30 23:53:50 |
| maintainer | |
| docs_url | None |
| author | |
| requires_python | >=3.7 |
| license | |
| keywords |
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| VCS |
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| requirements |
numdifftools
numpy
SciPy
sympy
tensorflow
|
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## chiku
### Efficient Probabilistic Polynomial Function Approximation Python Library.
#### Installation
To install run: <code>pip install chiku</code>
#### Approximation Libraries
Complex (non-linear) functions like Sigmoid ( $\sigma(x)$ ) and Hyperbolic Tangent ( $\tanh{x}$ ) can be computed with Fully Homomorphic Encryption (FHE) in an encrypted domain using piecewise-linear functions (a linear approximation of $\sigma(x) = 0.5 + 0.25x$ can be derived from the first two terms of Taylor series $\frac{1}{2} + \frac{1}{4}x$ ) or polynomial approximations like Taylor, Pade, Chebyshev, Remez, and Fourier series. These deterministic approaches yield the same polynomial for the same function. In contrast, we propose to use Artificial Neural Network ( $ANN$ ) to derive the approximation polynomial probabilistically, where the coefficients are based on the initial weights and convergence of the $ANN$ model. Our scheme is publicly available here as an open-source Python package.
Library | Taylor | Fourier | Pade | Chebyshev | Remez | ANN
--------|--------|---------|------|-----------|-------|-----
[numpy](https://github.com/numpy/numpy)||||✔||
[scipy](https://github.com/scipy/scipy)|✔||✔|||
[mpmath](https://github.com/mpmath/mpmath)|✔|✔|✔|✔||
[chiku](https://github.com/devharsh/chiku)|✔|✔|✔|✔|✔|✔
The table above compares our library with other popular Python packages for numerical analysis. While the $mpmath$ library provides Taylor, Pade, Fourier, and Chebyshev approximations, a user has to transform the functions to suit the $mpmath$ datatypes (e.g., $mpf$ for real float and $mpc$ for complex values). In contrast, our library requires no modifications and can approximate arbitrary functions. Additionally, we provide Remez approximation along with the other methods supported by the $mpmath$.
#### ANN Approximation
While $ANN$ are known for their universal function approximation properties, they are often treated as a black box and used to calculate the output value. We propose to use a basic 3-layer perceptron consisting of an input layer, a hidden layer, and an output layer; both hidden and output layers have linear activations to generate the coefficients for an approximation polynomial of a given order. In this architecture, the input layer is dynamic, with the input nodes corresponding to the desired polynomial degrees. While having a variable number of hidden layers is possible, we fix it to a single layer with a single node to minimize the computation.
We show coefficient calculations for a third-order polynomial $d=3$ for a univariate function $f(x) = y$ for an input $x$, actual output $y$, and predicted output $y_{out}$.
Input layer weights are
$\{w_1, w_2, \ldots, w_d\} = \{w_1, w_2, w_3\} = \{x, x^2, x^3\}$
and biases are $\{b_1, b_2, b_3\} = b_h$. Thus the output of the hidden layer is
$y_h = w_1 x + w_2 x^2 + w_3 x^3 + b_h$
The predicted output is calculated by
$y_{out} = w_{out} \cdot y_h + b_{out}$
$= w_1 w_{out} x + w_2 w_{out} x^2 + w_3 w_{out} x^3 + (b_h w_{out} + b_{out})$
where the layer weights $\{w_1 w_{out}, w_2 w_{out}, w_3 w_{out}\}$ are the coefficients for the approximating polynomial of order-3
and the constant term is $b_h w_{out} + b_{out}$.
Our polynomial approximation approach using $ANN$ can generate polynomials with specified degrees. E.g., a user can generate a complete third-order polynomial for $\sin(x)$, which yields a polynomial
$-0.0931199x^3 - 0.001205849x^2 + 0.85615075x + 0.0009873845$
in the interval $[-\pi,\pi]$. Meanwhile, a user may want to optimize the above polynomial by eliminating the coefficients for $x^2$ to reduce costly multiplications in FHE, which yields the following:
$-0.09340597x^3 + 0.8596622x + 0.0005142888.$
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"description": "## chiku\n### Efficient Probabilistic Polynomial Function Approximation Python Library.\n\n#### Installation\nTo install run: <code>pip install chiku</code>\n\n#### Approximation Libraries\nComplex (non-linear) functions like Sigmoid ( $\\sigma(x)$ ) and Hyperbolic Tangent ( $\\tanh{x}$ ) can be computed with Fully Homomorphic Encryption (FHE) in an encrypted domain using piecewise-linear functions (a linear approximation of $\\sigma(x) = 0.5 + 0.25x$ can be derived from the first two terms of Taylor series $\\frac{1}{2} + \\frac{1}{4}x$ ) or polynomial approximations like Taylor, Pade, Chebyshev, Remez, and Fourier series. These deterministic approaches yield the same polynomial for the same function. In contrast, we propose to use Artificial Neural Network ( $ANN$ ) to derive the approximation polynomial probabilistically, where the coefficients are based on the initial weights and convergence of the $ANN$ model. Our scheme is publicly available here as an open-source Python package.\n\nLibrary | Taylor | Fourier | Pade | Chebyshev | Remez | ANN\n--------|--------|---------|------|-----------|-------|-----\n[numpy](https://github.com/numpy/numpy)||||\u2714||\n[scipy](https://github.com/scipy/scipy)|\u2714||\u2714|||\n[mpmath](https://github.com/mpmath/mpmath)|\u2714|\u2714|\u2714|\u2714||\n[chiku](https://github.com/devharsh/chiku)|\u2714|\u2714|\u2714|\u2714|\u2714|\u2714\n\nThe table above compares our library with other popular Python packages for numerical analysis. While the $mpmath$ library provides Taylor, Pade, Fourier, and Chebyshev approximations, a user has to transform the functions to suit the $mpmath$ datatypes (e.g., $mpf$ for real float and $mpc$ for complex values). In contrast, our library requires no modifications and can approximate arbitrary functions. Additionally, we provide Remez approximation along with the other methods supported by the $mpmath$.\n\n#### ANN Approximation\nWhile $ANN$ are known for their universal function approximation properties, they are often treated as a black box and used to calculate the output value. We propose to use a basic 3-layer perceptron consisting of an input layer, a hidden layer, and an output layer; both hidden and output layers have linear activations to generate the coefficients for an approximation polynomial of a given order. In this architecture, the input layer is dynamic, with the input nodes corresponding to the desired polynomial degrees. While having a variable number of hidden layers is possible, we fix it to a single layer with a single node to minimize the computation.\n\n![Polynomial approximation using ANN](https://github.com/devharsh/chiku/blob/main/ANN-approximation.drawio.png \"Polynomial approximation using ANN\")\n\nWe show coefficient calculations for a third-order polynomial $d=3$ for a univariate function $f(x) = y$ for an input $x$, actual output $y$, and predicted output $y_{out}$.\n\nInput layer weights are\n\n$\\{w_1, w_2, \\ldots, w_d\\} = \\{w_1, w_2, w_3\\} = \\{x, x^2, x^3\\}$\n\nand biases are $\\{b_1, b_2, b_3\\} = b_h$. Thus the output of the hidden layer is\n\n$y_h = w_1 x + w_2 x^2 + w_3 x^3 + b_h$\n\nThe predicted output is calculated by\n\n$y_{out} = w_{out} \\cdot y_h + b_{out}$\n$= w_1 w_{out} x + w_2 w_{out} x^2 + w_3 w_{out} x^3 + (b_h w_{out} + b_{out})$\n\nwhere the layer weights $\\{w_1 w_{out}, w_2 w_{out}, w_3 w_{out}\\}$ are the coefficients for the approximating polynomial of order-3\n\nand the constant term is $b_h w_{out} + b_{out}$.\n\nOur polynomial approximation approach using $ANN$ can generate polynomials with specified degrees. E.g., a user can generate a complete third-order polynomial for $\\sin(x)$, which yields a polynomial\n$-0.0931199x^3 - 0.001205849x^2 + 0.85615075x + 0.0009873845$\nin the interval $[-\\pi,\\pi]$. Meanwhile, a user may want to optimize the above polynomial by eliminating the coefficients for $x^2$ to reduce costly multiplications in FHE, which yields the following:\n$-0.09340597x^3 + 0.8596622x + 0.0005142888.$\n",
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