Elementary Transcentental Functions

These types of activation functions where highly explored by Taehwan Kim and Tulay Adali, mainly in [CIT2001-KIM] and [CIT2003-KIM]. Please refer to them for further information. These functions are divided into 4 groups.

  • Circular
  • Inverse Circular
  • Hyperbolic
  • Inverse Hyperbolic

Circular

etf_circular_tan(z):

Computes tan of z element-wise.

\[tan(z) = \frac{e^{iz} - e^{-iz}}{i(e^{iz} + e^{-iz})}\]
Parameters:z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.
etf_circular_sin(z):

Computes sine of z element-wise.

\[sin(z) = \frac{e^{iz} - e^{-iz}}{2i}\]
Parameters:z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.

Inverse Circular

etf_inv_circular_atan(z):

Computes the trignometric inverse tangent of z element-wise.

\[atan(z) = \int_{0}^{z} \frac{dt}{1+t^2}\]
Parameters:z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.
etf_inv_circular_asin(z):

Computes the trignometric inverse sine of z element-wise.

\[asin(z) = \int_{0}^{z} \frac{dt}{(1-t)^1/2}\]
Parameters:z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.
etf_inv_circular_acos(z):

Computes acos of z element-wise.

\[acos(z) = \int_{z}^{1} \frac{dt}{(1-t^2)^1/2}\]
Parameters:z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.

Hyperbolic

etf_circular_tanh(z):

Computes hyperbolic tangent of z element-wise.

\[tanh(z) = \frac{sinh(z)}{cosh(z)} = \frac{e^{z} - e^{-z}}{e^{z} + e^{-z}}\]
Parameters:z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.
etf_circular_sinh(z):

Computes hyperbolic sine of z element-wise.

\[sinh(z) = \frac{e^{z} - e^{-z}}{2}\]
Parameters:z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.

Inverse Hyperbolic

etf_inv_circular_atanh(z):

Computes inverse hyperbolic tangent of z element-wise.

\[atanh(z) = \int_{0}^{z} \frac{dt}{1-t^2}\]
Parameters:z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.
etf_inv_circular_asinh(z):

Computes inverse hyperbolic sine of z element-wise.

\[asinh(z) = \int_{0}^{z} \frac{dt}{(1+t^2)^1/2}\]
Parameters:z – A Tensor. Must be one of the following types: bfloat16, half, float32, float64, int8, int16, int32, int64, complex64, complex128.
[CIT2001-KIM]
  1. Kim and T Adali “Complex Backpropagation neural network using elementary transdencental activation functions” 2001 IEEE International Conference on Acoustics, Speech, and Signal Processing. Proceedings (Cat. No.01CH37221)
[CIT2003-KIM]
  1. Kim and T Adali “Approximation by Fully Complex MLP Using Elementary Transcendental Activation Functions” 2001 Neural Computation