If they were interested in a fair comparison they would use a test set from (for example) Rubi or one of the CAS. My guess is that they did this: 1. generate a random expression S favoring + and * in the tree. 2. differentiate S to get S' 3. "learn" the integral of S'.
Here's the trick. S' will, with very high probability, be a sum. Say s1+s2+s3. A CAS will usually try to compute integrate(s1,x) + integrate(s2,x)+ integrate(s3,x). That's the way integral tables work too. Unfortunately, for many "random" expressions, s1, s2, s3, ... are NOT integrable in terms of elementary functions. Only their sum. So a CAS will fail. Here's a particular example. exp(-x^3)/x^4. Differentiate (I'm copy/pasting from Maxima) to get -(3*%e^(-x^3))/x^2- (4*%e^(-x^3))/x^5 Neither of these terms is separately integrable in terms of elementary functions. So a "real" CAS will fail on even "simple" problems. If you generate trees with 15 random operators, the probably of failing increases. For this particular example, which is the 2nd one I tried, Maxima gives gamma_incomplete(-1/3,x^3)+(4*gamma_incomplete(-4/3,x^3))/3 Non elementary it seems. But we know this is supposed to be the same as exp(-x^3)/x^4. A minute testing numerically suggests it is, indeed, equal. So what we have for "ML" here is a made-up test set that is not reflective of the actual task of computing integrals as needed in applied math, and as considered in (for example) integral tables or integration algorithms. We are perhaps familiar with the notion of "teaching for the test" in which students and teachers collude to get excellent grades on some standardized test. Yet the students may really not know the material. This is maybe worse because the "test" is not some important standardized suite of integration problems. It is just randomly generated. Maybe it would be fair to call it noise? The author could post the test suite, I suppose. RJF RJF On Tuesday, October 1, 2019 at 11:22:51 PM UTC-7, Emmanuel Charpentier wrote: > > > > Le mercredi 2 octobre 2019 01:48:15 UTC+2, rjf a écrit : >> >> I think that if you read the paper you would not expect it to compete >> with a CAS >> except on its made-up artificial testset. >> > > Could you amplify ? > > >> RJF >> >> >> On Monday, September 30, 2019 at 10:57:44 AM UTC-4, Martin R wrote: >>> >>> Actually, I think it would be even more interesting to compare with >>> FriCAS, because FriCAS has the most complete implementation of the Risch >>> algorithm and does not at all rely on pattern matching. >>> >>> Martin >>> >>> Am Sonntag, 29. September 2019 15:00:01 UTC+2 schrieb mmarco: >>>> >>>> I would be very interested in comparing their results with RUBI. >>>> >>>> El viernes, 27 de septiembre de 2019, 21:53:00 (UTC+2), Eric >>>> Gourgoulhon escribió: >>>>> >>>>> Thanks for sharing! >>>>> This looks very promising. I hope we have it in Sage some day. >>>>> >>>>> Eric. >>>>> >>>>> Le vendredi 27 septembre 2019 17:06:31 UTC+2, Dima Pasechnik a écrit : >>>>>> >>>>>> https://openreview.net/pdf?id=S1eZYeHFDS >>>>>> >>>>>> I wish they had code available... >>>>>> >>>>> -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/sage-devel/3349b278-f7cb-4ecb-bb19-233a9ab72928%40googlegroups.com.
