#approximationtheory — Public Fediverse posts
Live and recent posts from across the Fediverse tagged #approximationtheory, aggregated by home.social.
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I recently read two interesting survey articles by my academic brother Ben Adcock at Simon Fraser University about theoretical aspect of sampling: how to approximate a function 𝑓 given random point samples 𝑓(𝑥ᵢ) with noise. This is a fundamental problem in Machine Learning.
The first paper, "Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks" (by Ben and co-authors), is about how fast the approximation error may decrease as you take more samples. We can overcome the curse of dimensionality if the function gets increasingly smooth in higher dimensions. URL: https://arxiv.org/abs/2404.03761
The second paper, "Optimal sampling for least-squares approximation", is about choosing where to sample in order to get as close to the unknown function (in least-square sense) as possible. https://arxiv.org/abs/2409.02342
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I recently read two interesting survey articles by my academic brother Ben Adcock at Simon Fraser University about theoretical aspect of sampling: how to approximate a function 𝑓 given random point samples 𝑓(𝑥ᵢ) with noise. This is a fundamental problem in Machine Learning.
The first paper, "Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks" (by Ben and co-authors), is about how fast the approximation error may decrease as you take more samples. We can overcome the curse of dimensionality if the function gets increasingly smooth in higher dimensions. URL: https://arxiv.org/abs/2404.03761
The second paper, "Optimal sampling for least-squares approximation", is about choosing where to sample in order to get as close to the unknown function (in least-square sense) as possible. https://arxiv.org/abs/2409.02342
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I recently read two interesting survey articles by my academic brother Ben Adcock at Simon Fraser University about theoretical aspect of sampling: how to approximate a function 𝑓 given random point samples 𝑓(𝑥ᵢ) with noise. This is a fundamental problem in Machine Learning.
The first paper, "Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks" (by Ben and co-authors), is about how fast the approximation error may decrease as you take more samples. We can overcome the curse of dimensionality if the function gets increasingly smooth in higher dimensions. URL: https://arxiv.org/abs/2404.03761
The second paper, "Optimal sampling for least-squares approximation", is about choosing where to sample in order to get as close to the unknown function (in least-square sense) as possible. https://arxiv.org/abs/2409.02342
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I recently read two interesting survey articles by my academic brother Ben Adcock at Simon Fraser University about theoretical aspect of sampling: how to approximate a function 𝑓 given random point samples 𝑓(𝑥ᵢ) with noise. This is a fundamental problem in Machine Learning.
The first paper, "Learning smooth functions in high dimensions: from sparse polynomials to deep neural networks" (by Ben and co-authors), is about how fast the approximation error may decrease as you take more samples. We can overcome the curse of dimensionality if the function gets increasingly smooth in higher dimensions. URL: https://arxiv.org/abs/2404.03761
The second paper, "Optimal sampling for least-squares approximation", is about choosing where to sample in order to get as close to the unknown function (in least-square sense) as possible. https://arxiv.org/abs/2409.02342
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This chocolate reminded me of Bernstein ellipses, which govern how fast the Chebyshev approximation converges to a function.
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This chocolate reminded me of Bernstein ellipses, which govern how fast the Chebyshev approximation converges to a function.
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This chocolate reminded me of Bernstein ellipses, which govern how fast the Chebyshev approximation converges to a function.
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This chocolate reminded me of Bernstein ellipses, which govern how fast the Chebyshev approximation converges to a function.
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This chocolate reminded me of Bernstein ellipses, which govern how fast the Chebyshev approximation converges to a function.
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My thoughts keep turning back to the OWNA (One World Numerical Analysis) talk of Daan Huybrechs a few weeks ago. Most of numerical analysis is built on approximating functions in finite-dimensional spaces: \[ f(x) \approx \sum_i a_i \varphi_i(i), \] where 𝑓 is the function we want to approximate and φᵢ are easy functions like polynomials. In the standard setting, the φᵢ form a basis. The talk explained why you sometimes want to add some more "basis" functions, which destroys the linear independence of the φᵢ so that they are no longer a basis. The main topic was the theory behind this.
As motivation, consider the square root function on [0, 1]. This is not analytic at x=0 and approximation by polynomials does not converge fast. However, you can get fast convergence (root exponential IIRC) if you use rational functions. More generally, the solution of Laplace's equation on a domain with re-entrant corners has singularities at the corners. The lightning method uses an overcomplete "basis" of polynomials and rational functions, which converges fast.
It's one of those talks that I wished I understood fully, but it would take me over a month of sustained effort or more to do so. Hopefully I will find an excuse to immerse myself in the topic.
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My thoughts keep turning back to the OWNA (One World Numerical Analysis) talk of Daan Huybrechs a few weeks ago. Most of numerical analysis is built on approximating functions in finite-dimensional spaces: \[ f(x) \approx \sum_i a_i \varphi_i(i), \] where 𝑓 is the function we want to approximate and φᵢ are easy functions like polynomials. In the standard setting, the φᵢ form a basis. The talk explained why you sometimes want to add some more "basis" functions, which destroys the linear independence of the φᵢ so that they are no longer a basis. The main topic was the theory behind this.
As motivation, consider the square root function on [0, 1]. This is not analytic at x=0 and approximation by polynomials does not converge fast. However, you can get fast convergence (root exponential IIRC) if you use rational functions. More generally, the solution of Laplace's equation on a domain with re-entrant corners has singularities at the corners. The lightning method uses an overcomplete "basis" of polynomials and rational functions, which converges fast.
It's one of those talks that I wished I understood fully, but it would take me over a month of sustained effort or more to do so. Hopefully I will find an excuse to immerse myself in the topic.
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My thoughts keep turning back to the OWNA (One World Numerical Analysis) talk of Daan Huybrechs a few weeks ago. Most of numerical analysis is built on approximating functions in finite-dimensional spaces: \[ f(x) \approx \sum_i a_i \varphi_i(i), \] where 𝑓 is the function we want to approximate and φᵢ are easy functions like polynomials. In the standard setting, the φᵢ form a basis. The talk explained why you sometimes want to add some more "basis" functions, which destroys the linear independence of the φᵢ so that they are no longer a basis. The main topic was the theory behind this.
As motivation, consider the square root function on [0, 1]. This is not analytic at x=0 and approximation by polynomials does not converge fast. However, you can get fast convergence (root exponential IIRC) if you use rational functions. More generally, the solution of Laplace's equation on a domain with re-entrant corners has singularities at the corners. The lightning method uses an overcomplete "basis" of polynomials and rational functions, which converges fast.
It's one of those talks that I wished I understood fully, but it would take me over a month of sustained effort or more to do so. Hopefully I will find an excuse to immerse myself in the topic.
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The answer is a definite “yes.” Hamming named his E as “the shift operator”. If we read Powell’s proof on the convergence (Th. 6.3) and smoothness (Th. 6.4) carefully, and follow the smoothness proof to higher derivatives, we see that the use of binomials (clearly emphasized in Hamming), involves rather tedious shifting of summation indices.
Hamming obviously the engineer, eliding the extra work as part of a subroutine!
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The answer is a definite “yes.” Hamming named his E as “the shift operator”. If we read Powell’s proof on the convergence (Th. 6.3) and smoothness (Th. 6.4) carefully, and follow the smoothness proof to higher derivatives, we see that the use of binomials (clearly emphasized in Hamming), involves rather tedious shifting of summation indices.
Hamming obviously the engineer, eliding the extra work as part of a subroutine!
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it seems to me that the operator equation given by Hamming (1989) Ch6 Sec7 is closely related, if not the same as, the Bernstein operator shown in Powell (1981) Eq 6.23
Can someone familiar with #ApproximationTheory or #NumericalAnalysis confirm?
Dover did a nice job reprinting Hamming’s book, but I wish they preserved or appended a citation listing. There’s no bibliography or reference index at all!
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it seems to me that the operator equation given by Hamming (1989) Ch6 Sec7 is closely related, if not the same as, the Bernstein operator shown in Powell (1981) Eq 6.23
Can someone familiar with #ApproximationTheory or #NumericalAnalysis confirm?
Dover did a nice job reprinting Hamming’s book, but I wish they preserved or appended a citation listing. There’s no bibliography or reference index at all!
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here's the proof that engendered my complaint. Powell uses Bolanzo-Weierstrauss to prove a fixed point theorem for a sequence of operators. This gives support for an earlier theorem establishing a boundary condition on least errors for approximating functions. #ApproximationTheory
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here's the proof that engendered my complaint. Powell uses Bolanzo-Weierstrauss to prove a fixed point theorem for a sequence of operators. This gives support for an earlier theorem establishing a boundary condition on least errors for approximating functions. #ApproximationTheory
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CW: Approximation Theory - Motivating Concepts
#ApproximationTheory and #NumericalMethods constitute the arts and sciences of acquiring "close enough" calculations for computationally intractable functions.
Let's unpack this
"computationally intractable" simply means it's either inefficient or impossible to obtain an exact answer to a function - either mechanically or in general.
Take root two as an example. In some sense, root two is exact. But a mechanical representation is not possible
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CW: Approximation Theory - Motivating Concepts
#ApproximationTheory and #NumericalMethods constitute the arts and sciences of acquiring "close enough" calculations for computationally intractable functions.
Let's unpack this
"computationally intractable" simply means it's either inefficient or impossible to obtain an exact answer to a function - either mechanically or in general.
Take root two as an example. In some sense, root two is exact. But a mechanical representation is not possible
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I've been studying the past few weeks #ApproximationTheory through #OpenUniversity -- an excellent and affordable distance learning institution for English speakers. This topic I've written about before ( see https://mathstodon.xyz/@jared/109313785067280336 ); and I'd like to continue my slow approach to understanding by sharing my personal learning experience here.
So as not to take too much space on folks timeline, I'm going to update this thread with a "Approximation Theory" CW. But, fear not, this is fun stuff!
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I've been studying the past few weeks #ApproximationTheory through #OpenUniversity -- an excellent and affordable distance learning institution for English speakers. This topic I've written about before ( see https://mathstodon.xyz/@jared/109313785067280336 ); and I'd like to continue my slow approach to understanding by sharing my personal learning experience here.
So as not to take too much space on folks timeline, I'm going to update this thread with a "Approximation Theory" CW. But, fear not, this is fun stuff!