Polynomial approximation of a function in a chosen interval
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I don't know if this should be in the math forum or the computer science one, but I think it is more relevant in the math section.
I would like to approximate nonlinear functions typically used in neural networks on a given interval. However, the Taylor polynomial approximations are in a neighborhood of a point, leading to overflows in my computations. What's more, I don't think Taylor polynomials are the best ones to used, due to the degree required to fit "well" the function to approximate.
Do you have other algorithms than Chebyshev / Taylor polynomials to approximate functions, that might be faster / more accurate to fit a given function in an interval [a;b]?
Thank you.
polynomials approximation approximation-theory
asked Dec 14 '18 at 6:44
Robin S.Robin S.
63
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add a comment |
$begingroup$
I don't know if this should be in the math forum or the computer science one, but I think it is more relevant in the math section.
I would like to approximate nonlinear functions typically used in neural networks on a given interval. However, the Taylor polynomial approximations are in a neighborhood of a point, leading to overflows in my computations. What's more, I don't think Taylor polynomials are the best ones to used, due to the degree required to fit "well" the function to approximate.
Do you have other algorithms than Chebyshev / Taylor polynomials to approximate functions, that might be faster / more accurate to fit a given function in an interval [a;b]?
Thank you.
polynomials approximation approximation-theory
asked Dec 14 '18 at 6:44
Robin S.Robin S.
63
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$begingroup$
If you are given a number of data points on the polynomial you want to approximate, interpolation would probably work best. Or at least, it's the first to come to mind.
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– Eevee Trainer
Dec 14 '18 at 6:47
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Did you think about Padé approximants ? Could you provide a function and a interval ?
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– Claude Leibovici
Dec 14 '18 at 8:10
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It's for neural networks, so basically the activation functions are either sigmoid or tanh, and as an interval we could take [-8,8] for example.. hmmm Padé approximants don't give you polynomial solutions, and I need polynomials as they are only defined in term of additions and multiplications
$endgroup$
– Robin S.
Dec 15 '18 at 4:11
add a comment |
$begingroup$
I don't know if this should be in the math forum or the computer science one, but I think it is more relevant in the math section.
I would like to approximate nonlinear functions typically used in neural networks on a given interval. However, the Taylor polynomial approximations are in a neighborhood of a point, leading to overflows in my computations. What's more, I don't think Taylor polynomials are the best ones to used, due to the degree required to fit "well" the function to approximate.
Do you have other algorithms than Chebyshev / Taylor polynomials to approximate functions, that might be faster / more accurate to fit a given function in an interval [a;b]?
Thank you.
polynomials approximation approximation-theory
asked Dec 14 '18 at 6:44
Robin S.Robin S.
63
$endgroup$
I don't know if this should be in the math forum or the computer science one, but I think it is more relevant in the math section.
I would like to approximate nonlinear functions typically used in neural networks on a given interval. However, the Taylor polynomial approximations are in a neighborhood of a point, leading to overflows in my computations. What's more, I don't think Taylor polynomials are the best ones to used, due to the degree required to fit "well" the function to approximate.
Do you have other algorithms than Chebyshev / Taylor polynomials to approximate functions, that might be faster / more accurate to fit a given function in an interval [a;b]?
Thank you.
polynomials approximation approximation-theory
polynomials approximation approximation-theory
asked Dec 14 '18 at 6:44
Robin S.Robin S.
63
asked Dec 14 '18 at 6:44
Robin S.Robin S.
63
asked Dec 14 '18 at 6:44
Robin S.Robin S.
63
asked Dec 14 '18 at 6:44
Robin S.Robin S.
63
asked Dec 14 '18 at 6:44
Robin S.Robin S.
63
63
$begingroup$
If you are given a number of data points on the polynomial you want to approximate, interpolation would probably work best. Or at least, it's the first to come to mind.
$endgroup$
– Eevee Trainer
Dec 14 '18 at 6:47
$begingroup$
Did you think about Padé approximants ? Could you provide a function and a interval ?
$endgroup$
– Claude Leibovici
Dec 14 '18 at 8:10
$begingroup$
It's for neural networks, so basically the activation functions are either sigmoid or tanh, and as an interval we could take [-8,8] for example.. hmmm Padé approximants don't give you polynomial solutions, and I need polynomials as they are only defined in term of additions and multiplications
$endgroup$
– Robin S.
Dec 15 '18 at 4:11
add a comment |
$begingroup$
If you are given a number of data points on the polynomial you want to approximate, interpolation would probably work best. Or at least, it's the first to come to mind.
$endgroup$
– Eevee Trainer
Dec 14 '18 at 6:47
$begingroup$
Did you think about Padé approximants ? Could you provide a function and a interval ?
$endgroup$
– Claude Leibovici
Dec 14 '18 at 8:10
$begingroup$
It's for neural networks, so basically the activation functions are either sigmoid or tanh, and as an interval we could take [-8,8] for example.. hmmm Padé approximants don't give you polynomial solutions, and I need polynomials as they are only defined in term of additions and multiplications
$endgroup$
– Robin S.
Dec 15 '18 at 4:11
$begingroup$
If you are given a number of data points on the polynomial you want to approximate, interpolation would probably work best. Or at least, it's the first to come to mind.
$endgroup$
– Eevee Trainer
Dec 14 '18 at 6:47
$begingroup$
If you are given a number of data points on the polynomial you want to approximate, interpolation would probably work best. Or at least, it's the first to come to mind.
$endgroup$
– Eevee Trainer
Dec 14 '18 at 6:47
$begingroup$
Did you think about Padé approximants ? Could you provide a function and a interval ?
$endgroup$
– Claude Leibovici
Dec 14 '18 at 8:10
$begingroup$
Did you think about Padé approximants ? Could you provide a function and a interval ?
$endgroup$
– Claude Leibovici
Dec 14 '18 at 8:10
$begingroup$
It's for neural networks, so basically the activation functions are either sigmoid or tanh, and as an interval we could take [-8,8] for example.. hmmm Padé approximants don't give you polynomial solutions, and I need polynomials as they are only defined in term of additions and multiplications
$endgroup$
– Robin S.
Dec 15 '18 at 4:11
$begingroup$
It's for neural networks, so basically the activation functions are either sigmoid or tanh, and as an interval we could take [-8,8] for example.. hmmm Padé approximants don't give you polynomial solutions, and I need polynomials as they are only defined in term of additions and multiplications
$endgroup$
– Robin S.
Dec 15 '18 at 4:11
add a comment |
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$begingroup$
If you are given a number of data points on the polynomial you want to approximate, interpolation would probably work best. Or at least, it's the first to come to mind.
$endgroup$
– Eevee Trainer
Dec 14 '18 at 6:47
$begingroup$
Did you think about Padé approximants ? Could you provide a function and a interval ?
$endgroup$
– Claude Leibovici
Dec 14 '18 at 8:10
$begingroup$
It's for neural networks, so basically the activation functions are either sigmoid or tanh, and as an interval we could take [-8,8] for example.. hmmm Padé approximants don't give you polynomial solutions, and I need polynomials as they are only defined in term of additions and multiplications
$endgroup$
– Robin S.
Dec 15 '18 at 4:11