Fitting manifolds to data - Charlie Fefferman

it's it's so it's it's a good thing that maybe that the task is is to find something to say about each speaker not available through Google rather than say Cambridge analytic oh okay all right okay I frankly wondered what I would say and and the talk that you are about to hear is is based on the on a kind of minimax approach in which I hope that no one will be utterly bored by absolutely everything I say okay so I want to talk about two problems of fitting fitting manifolds to data there is an intrinsic flavor and there is an extrinsic flavor and both of these tie in although all right so I am I mean you can see the evidence is on Google for example I am a joint author of the papers that I will be speaking on but I'm not the main contributor to the papers that I will be speaking on but these papers relate to something that that I have been studying for a very long time and so that is a third flavor of the problem which which maybe I will discuss a little all right so let me start with the extrinsic flavor so we are in some very high dimensional Euclidean space RN it might be Hilbert space but let's let's just keep it are in distances are always just the Euclidean distance okay for this problem now we we are given a large collection of data points let's say X 1 through X s s is the sample size okay these points lie in RN they are sampled independently at random from some completely unknown probability distribution which I will call DP so this is an unknown probability distribution on our okay and so these are sampled independently from that and based based on viewing these we would like to test the hypotheses we'd like to test the hypothesis that this probability distribution is concentrated close to a low dimensional embedded manifold with reasonable geometry okay why do we want to do that well people who study the real world say that this is a useful hypothesis I certainly don't know anything about the real world but but this seems like an interesting problem so so we want to test the manifold hypothesis so I want to test the hypothesis that there exists an embedded manifold of some lower dimension let's say D sitting inside RN with reasonable geometry such that deep this probability distribution is concentrated close to him okay now so that's that's the extrinsic problem of course this problem is not well formulated what does reasonable geometry mean what does concentrated close to em mean and for that matter what does it mean to test the hypothesis okay well first of all let's let's do the easy thing first what does it mean to say that P is concentrated close so let's say I'm going to take the expected value under the probability distribution DP of the of the squared distance to the manifold so I take I take a point X which is sampled according to DP and I take the distance squared from X to M and I would like to say that the the expectation of that is less than Epsilon and so let's say I specify epsilon and that will be that's what I mean by concentrated close I would love to be able to deal with a particular epsilon but there will be some slop maybe in the good case I will be able to achieve a hundred epsilon and in the bad case I will know that I cannot achieve epsilon over a hundred or something like that okay immediately this this is subject to trouble for the following reason I want to make no assumptions on DP the only thing I know about DP is that is that these points are sampled from it that's all the information I have imagine that DP has a tiny tiny tiny probability much smaller than one over s that the that that the point X is out in the Andromeda galaxy someplace in that case this expected value will be a small probability multiplied by a huge number and I will be out of luck and I will fail to see that so so it is impossible to test something like this on lesson make some assumptions and so what I'm going to assume is that this is not this doesn't live on all of our n but lives on the unit ball the ball of radius one about zero about the origin in RN so this number one is a unit distance that's that's as far away as Oh capital at capital n is arbitrarily large and it's very important that constant it's it's varying various things are supposed to be independent of n the fact the fact that this is one instead of some letter of the alphabet is of course irrelevant because I can just scale my data but it means that various formulas that look dimensionally incorrect are actually perfectly alright because the Union I mean that specifies what a length means okay so all right I would also like to say then that this M is contained in the same ball v01 okay so all right I've now fixed the the trivial fatal error and so this maybe I'll put a symbol like that indicating that there's some slop in the epsilon but the slop is not dependent on capital n capital n could be infinity without loss of generality I could be working in Hilbert space yes right that's correct that's correct no no no no one way to think of it is P can be absolutely arbitrary but if I see any data points from the Andromeda galaxy I will ignore them as outliers and and they will not count in this that's right Oh as long I'm eeen you can you can make you you can relax that assumption that's that's your question I think you can relax that assumption okay so so this is what it means concentrated close what does reasonable geometry mean well reasonable geometry of course is in the eye of the holder here's what it's going to mean for purposes of the extrinsic problem that I will talk about so reasonable geometry means two things first of all all right so M is of dimension D so it has a D dimensional volume and I'm going to specify a bound for it and again this this bound is subject to slop maybe in in the K in the good case in which the data really do like close to something here this volume will be less than a hundred times V and in the bad case I will guarantee that it cannot be done with volume less than V over 100 something like that but the constant 100 which in fact will be a much worse constant than 100 none of this is actually practical the the implied constants here again are independent of capital n capital n again could be infinity we could be working in Hilbert space okay so that's one condition the other is that the reach of M is greater than or equal to some number Tao now the reach of M is is a distance and I don't know how well-known this this concept is when I when I first became involved in the production of this paper I didn't know what it meant let me describe it and tell you what it means morally alright so so here's here's a manifold m and we look at a point not necessarily on the manifold but not too far from it if if this point is of distance less than tau from the manifold then there must be a unique closest point of the manifold to that point that's what reach means it's the the largest number tau for which that property is true okay what does that mean let's give some examples well first there's there's an infinitesimal condition a local purely local condition for the reach which is that all the sectional curvatures are less than tau I think of you know think think of a little tiny circle or think of think of a parabola with you know with enormous curvature at the tip and if you are if you are right there you are very close to the manifold but there is not a unique closest point of the manifold so so having a lower bound for the reach tells you a local condition on the curvatures but there's also a global aspect to it because for example if you see a spiral locally did I say the sectional curvatures I'm sorry the principle curvatures this is embedded I'm sorry I mean here you know the principal curvatures are very gentle but this has a lousy reach because this point is equally close to these two so we're going to assume a lower bound for the reach and and so these these are our conditions okay now what does it mean what mean to test to test the manifold so the manifold hypothesis is going to be that there exists an embedded D dimensional manifold D is given think of D as being three or five okay nice and small whereas n is enormous so we're going to test the hypothesis that there is such a manifold having these properties in such a way that this is true okay that's what we're supposed to test now how do we test it well we so we we are supposed to produce an algorithm and the algorithm takes some inputs and and it produces some outcome the inputs to the algorithm are of the parameters epsilon V tau how about D let's let's input D okay the dimension and one other parameter Delta which I haven't mentioned yet Delta is going to be a bound for the probability that the algorithm makes a mistake okay so those are the inputs and X 1 through X s in our RN where s is big enough s is greater than or equal to s min of these parameters D epsilon V tau and Delta okay where this fellow should be computable and in fact when the paper gives bounds I'll come back to that so those are the inputs okay so the parameters of the problem and enough sample points what are the output what are the out the outcomes there are two possible outcomes depending on the data one outcome is that the computer says no go sorry I'm unable to find amount of fold in which case presumably there is no such manifold okay or let's see how many guess I should come over here or okay or a manifold M of dimension D with me satisfying those conditions so volume of M less than or equal to C times V C depends only on D reach of M greater than or equal to tau what little C times tau this little C also depends only on D and let's see what else so okay well it does that now ah what a good question okay so first of all this manifold is sits embedded in a in a relatively low dimensional space relevant again I'm not talking about anything practical but this sits inside a vector space of the enormous RN of dimension controlled in terms of these parameters and and then once it's in there then it's given by a finite number of equations which are guaranteed to determine a nice manifold because they're suitably transverse okay good enough ok so that's what it means that it gives manifold now now alright so far I haven't said anything useful because for example an example of an algorithm that does this is that the algorithm can always say no go thank you ok so we need to do somewhat better than that and so the property of this for this algorithm to be acceptable so let's say the algorithm so we we accept the algorithm if the following condition is true pick any DP unknown probability distribution there pick any one okay concentrated in the ball okay then alright also specify the inputs then let's see okay having specified all right so specify these parameters and then take these guys by sampling independently at random from from DP run the algorithm and see what happens with probability at least one minus Delta the following two things happen either the algorithm outputs no-go and there is no embedded manifold with parameters like this way that this is a small constitute that this is a big constant these change places okay either there is no manifold with with with these with the property that the expected squared distance is less than a small constant depending on only on little D times Epsilon that's the no-go case or the algorithm outputs this manifold and this manifold then is guaranteed to have these properties and furthermore has the property that then this expected squared distance is is less than Big C times epsilon if all that stuff is true then we accept the algorithm and that means that in principle we can solve the the problem of testing the manifold hypothesis well I mean it could be there's no reason why it should be but if you want to make it so these these constants have the property that any big see you know this it's the usual thing it the statement gets weaker as Big C gets gets bigger and so take all of your big C's and just take their maximum and erase all the big season and put them all there and similarly for little C if if I were trying to tune it so in other word to tune it there's no reason why any of these constants are the same that's correct that's correct all absolutely right okay okay thank you okay yeah okay but in fact in fact it will be it will be independent of capital in money oh what a good question these manifolds do not have bounds and that these are this is a compact manifold thank you I should have said this is going to be a compact manifold and so one could definitely ask one could definitely ask and in so in fact one of the one of the natural questions in this subject is is is this really what you want so for example for example suppose let's take a very degenerate case capital n is three we have imagined that we sample a lot of points which are in r3 which are concentrated very close to a mobius strip well there is no manifold in r3 that you know with good geometry etc that suggests that maybe we are not asking the right question on the other hand imagine a bunch of points on a sphere in r3 which lie in some cantor set of ours on the sphere on the unit sphere well we're not we don't have complete geometrical information about how these points are distributed but the fact that they all lie very on or very close to the unit sphere is worth cheering so so okay yeah locally it's great so so okay so I would say yes I agree with you what's wrong with a mobius strip but but the Mobius strip is not a manifold because it has it has an edge okay so so so therefore so peer was asking okay wait a minute should we allow a boundary and well yes maybe we should allow a boundary and then what kind of geometries all right I do not well so remember this is this is all right without all right strictly speaking yes but morally no okay so so if I mean my probability distribution is concentrated in the unit ball I said that my manifold is embedded in the unit ball but I should have said that it's embedded in a ball of radius 10 and there's no difference in the proof okay fair fair point okay yeah would if everything lies on a high part if everything lies on let's say a two dimensional plane we should allow that as an interesting solution okay I should say so so far I've only talked about the the extrinsic flavor I've given a bunch of definitions and no assertion but but I should I should say all of this is only the beginning of the subject these are the easiest questions one could ask I think well Assaf came up with with with an easier question what if what if the sample size is allowed to depend on n butBut I mean there's a hierarchy of questions of which these are just the the most the easiest and most elementary okay so there is there is a paper by Sanjoy Mitter let let me write the names down Sanjoy knitter harry narayan them and me called testing the manifold hypothesis okay and the the content of this paper is that there is an algorithm that that does this okay and once you think about it it's actually not so bad and and maybe I will have time to explain a little there are explicit estimates for how many operations you have to do you know how how much computation does the algorithm require it requires a ridiculous amount of computation it's very far from practical very far there is also an explicit expression that I think I won't write down for s men which has the feature that it is definitely not optimal because the other day are you a contour ovitch sent me an email saying that he has a better bound and I haven't had time to check it but I assume it's it's fine so so so whatever it says in our paper disregard it because it can be very likely improved okay that is the that is the extrinsic version of the problem and that and the statement again the statement of the result for the extrinsic problem is that there is an algorithm that does this that that is acceptable in the sense that I just described well okay so let's see if there is if there is a manifold right that's that's part of being acceptable if there is a manifold you will find a manifold that does the job with ninety five with let's with probability at least one minus Delta okay maybe I here maybe I stopped writing after I wrote we accept the algorithm but yes so so in order to be acceptable the algorithm has to have two properties if there is a manifold that does what we want then with probability at least one minus Delta we the algorithm will produce such a manifold okay on the other hand if there is no such manifold the algorithm will not spiritus an am it will instead say no go no no the dimension is fixed the per at me I mean the parameters like let's say the the the let's say the volume might be a thousand times more than we would like but only only a thousand times more some fixed constant okay and or the reach the you knows the curvatures it might be a thousand times more curved than we would have liked but but within about if you're willing to accept so to speak orders of magnitude then I I see puzzlement so so I am NOT being clear it's okay alright but but the dimensions absolutely fix care oh yes oh yes yes so here is a very simple example let let us imagine that the actual the actual manifold is the sphere but all the points lie in the southern hemisphere okay well then the the manifold hypothesis is true and the algorithm will output something which will probably look a lot like the southern hemisphere joined together somehow to make a manifold which may or may not look like the northern hemisphere thank you it's a important point okay all right all right in particular that's a significant part of what makes that I mean the problem would be a lot easier if we were guaranteed that DP produces points close to any given point of the of the manifold so so we have to work to guess part of the manifold why do we want part of the manifold where where there are no points well that's that's the problem as stated alright let's let's abandon the extrinsic flavor for the moment and let me talk about the intrinsic version of the problem okay yes yes oh that that's so that that's that's what it somehow that's what it means well inscribe a ball wait wait wait I think maybe maybe you and I are thinking too low dimensionally we're in a big hilbert space and we have a three-dimensional thing what does it mean to inscribe a ball so think there's a bit within a minute so there's a a tubular neighborhood of some around around the let's say the three dimensional manifold M there's a tubular neighborhood of all the points that are of distance less than tau from from there and and any point in that tubular neighborhood is closest to one point of it to a unique point of M okay so for example if they were curves in the plane then then you could say that that on either side of the curve you can you can inscribe a circle and that would mean the same thing okay okay you had a question oh that's a very that's a very reasonable question no we are not a simple thank you we're not assuming that the manifold is connected oh and so for example I should have said that back back when the paper was a little fresher in my mind I guess I would have known that but I forgot know the manifold need not be connected and indeed you know suppose that you have two spheres and you know yeah yeah no no it will it will it will catch that thank you pardon but I asked you to I asked it to be closed so for example two spheres you know know it compact no boundary perfectly nice manifold not connected okay any other any other additions Corrections okay let's do the extrinsic and the intrinsic flavours sorry okay okay so in the all right in the intrinsic flavor we don't have data points in our n our data points live in the metric space itself the only data we are given will be distances and we would like to think that these that that this metric space we have found lies inside a manifold now this gives rise then to a deep question - about which I know basically nothing and and I in the intrinsic results that that I tell you about will be just the baby case of the real problem so the real problem which is I think very tough is given a finite metric space can you embed it isometrically into a reasonable or with small distortion into a Romanian manifold with with reasonable geometry where you get to specify what reasonable geometry means okay in the baby case this finite metric space let's say the metric space is finite and let's say let's say that what we want about this this finite metric space is not that it not that it merely embeds but that it is isometric or maybe isometric with small errors I'll be more precise in a minute isometric to a fine net inside a manifold so a finite set of points of which we have picked enough that every point of the manifold lies very close to one of them for such a thing sort of if you if you look at the finite metric space and squint you will see the manifold okay so this is a much easier problem if in the for the general problem if you look at a finite metric space and squint you cannot tell what it may embed into alright so all right the so the paper that I'm describing now is called is called reconstruction and interpolation of manifolds 1 the geometric Whitney problem clearly a paper written by committee so I'll give you the committee in a moment okay so who who are the authors there is Sergey you've on/off Slava Korolev Harry never on let me see if I've there enough mop he lost us dear dear I'm so sorry Harry ok good question on the list of authors yes yes yes so I'm sorry so you're saying isn't this problem easy because is that what you are saying no well well all right let me let me say the the theorem and then we will see first of all is it indeed very easy well yes actually I would claim it is although details are you know as in anything you have to work out the details they're not entirely a walkover but is it fairly described by what I said about squinting well you decide all right so so in the paper or if the paper actually talks about a general metric space and about complete manifolds I'm going to talk about finite metric spaces and compact manifolds in this in this talk so suppose we have two two metric spaces X DX and y dy I'm going to say they're approximately isometric with parameters a de and Delta if the following occurs okay so let's say there's a map F from X to Y with the property that if you look at any two points x1 and x2 in X then if you look at their distances well the distances of between their images is distorted only a little bit from from there from there this is the distance in Y of course and I'll put here let's say 1 + ADA times the distance in X between x1 and x2 but I'll add an error Delta so that at very small scales I don't claim that they were approximately the same and similarly let's say I have here 1 minus a2 times the distance in X x1 x2 minus Delta and so this this or some something trivially modified from this is a very common thing which is used for example in geometric group theory although they're typically some of the constants are large and here the constants are supposed to be small okay so so we want that and we want also another map G taking Y to X with similar properties and these are not necessarily inverses to each other okay so two metric spaces are approximately isometric so for example if we have a very very fine net in in a manifold and if we take one of these metric spaces to be the fine net and the other space and the other to be the manifold itself they are approximately isometric okay so this Delta is a length scale below that length scale all bets are off and and and above that knot and much above that length scale distance or distances are distorted only by by a two percent okay so the problem is to decide given a metric space in particular finite metric space decide and in Pacific in particular decide by an algorithm whether there exists a manifold and if there is a manifold then explain it there that produce it with the property that that the given metric space is approximately isometric with given parameters and the manifold has has reasonable geometry okay now what does reasonable geometry mean okay reasonable geometry means reasonable geometry means two things first of all bound bounds on the absolute value of sectional curvatures and a lower bound for the injectivity radius yes from above and below no no no no the bounds it can be 0 it cannot be plus a billion and it cannot be - a billion but it can be 0 ok ok so the upper bounds and the absolute value yeah ok all right know what all right so let's just think about nice manifolds again I'm thinking of compact but one couldn't do I mean the paper does complete so what are the properties of a manifold with these properties I mean I'm sorry that's Wow Donald Trump has the best words but I seem not to have the best words ok given this kind of equivalence what can you say about what can you say about a manifold with reasonable geometry that survives this kind of equivalence that's there those are somewhat better words ok well the the ok one thing we can say is that at some length scale or other every ball looks like a ball in our in our n oh I forgot to say what the dimension of the manifold is oh well the manifold is of dimension n it's given and it's supposed to be not very big ok so all right there is some length scale are such that below is such that any ball of radius R looks very close in the in in this sense to a ball of radius R in Euclidean space that is something easy to decide by an algorithm okay maybe I will take two minutes and and explain that but not immediately okay so so that is that is the first that is the first obvious condition the second obvious condition is what Pierre said which is that if you have too far away points and you want to connect them they must be connected by something like a geodesic what is it geodesic what what survives about geodesics if you view them this way and the answer is the following here I should have erased this this is an extrinsic statement okay so so maybe I should say all right so first the ball in in the metric space X here this looks like the ball in RM 0 r up to some equivalence a de Delta okay and the other other thing that the property of geodesics is the following ok let X Prime and X double prime be any two points of X then there exists a finite sequence let's so let's say X prime is the first of these points then there's X 2 then there's X and let's say and that's the last one such that the following occurs let's see the distance between between each of these X I and the next is less than Delta so these are all small you know these are small jumps we can get there by small jumps and the second thing is that if I is less than J is less than K then the distance between X I and X K that's obviously less than the distance between X I and X J added to the distance from X J to X K thank you that's the triangle inequality but the the absolute value of this distance should of this difference should be less than Delta for all IJ K ok and if if things look yes if things look like let's see trying to think whether I've got this right it's been awhile I sincerely hope I have this right oh well yeah III will look it up in a minute in Czech but the claim is that that if if with suitable Aida and Delta if if things look like they look like a manifold then this condition is true okay and the theorem is for certain for for suitable ranges of Delta and and Aida and are that in fact if these conditions are true then there is a manifold with bounds on its sectional curvature and injectivity radius in such a way that this can be done but before I I'm sorry this is much too sloppy and I will I will take a look and write down actually what the bounds are but before I do let me just take a step back and ask why why should one care about the intrinsic version of the problem the extrinsic version of the problem is is some some abstract take on something which is actually used by statisticians in the real world in order to understand data where does this problem come from the it's not just it's not just fooling around for its own sake it actually comes from medical imaging imagine imagine that that you apply some technology to look inside your body and and you you look at some goo and this goo is is perhaps not isotropic and but it is rep the the non isotropy that whatever the conductivity or the whatever of of the goo inside of you is represented by a Romani and metric okay I'm not making that in the immortal words of Anna Russell I'm not making this up you know do do people know that Anna Russell routine Anna Russell is describing the story of the ring of the Nibelungen it begins in the Rhine River is she is a musical humorist okay so that our story begins in the Rhine River in it okay if it were New York it would be the Hudson okay it is it is a hilarious monologue I'm sure it's available on YouTube and at some point zigfried meets whom does he meet either either Brunhilde or gue true and I don't remember and and Anna Russell announces this is the the first woman zigfried who has ever met who is not his aunt I'm not making this up you know so when I speak of goo inside us represented by a Riemannian metric that's that's serious okay I'm not making this up that at worst someone else made it up okay in any event that's where the problem comes from all right now maybe I will look at my notes and produce a precise theorem and a precise algorithm okay all right so suppose we have a metric space XD let's say it's finite doesn't have to be but if I can make it finite suppose that it is Delta close to RN that's that's this condition where'd it go that's this condition at at scale or in other words the any ball of radius R looks like so it looks like ball in are in up to error Delta in the metric we don't there's not even in order just there that and we assume that X is Delta intrinsic which means this condition this is called Delta intrinsic okay and if that is true then there exists a manifold M G of dimension and such that our metric space is and now here are the alright Delta is the length scale I mean Delta is the yes and there is a constant depending only on n times Delta over R these are the parameters almost isometric M D and G and furthermore the sectional curvatures are less than or equal to what is it less than or equal to a constant Delta over R cubed and injectivity radius greater than R over two that's that's the theorem and and those conditions are sharp now what what do all these what do all these conditions mean let's take a simple example let's say let's say that we have a fine net in in the unit in the unit sphere in r3 okay okay so it's her it's sectional curvatures are of the order of magnitude of one you might wonder us suppose I want to find something which is for which Delta is very small well if Delta is very small then a ball I mean a ball in the unit sphere so a little spherical cap of radius R spherical radius R looks Delta close to a Euclidean ball if it turns out if if R is of the order of magnitude of Delta to the one-third work it out okay it's a little calculus problem and so I should think that R is like Delta to the one-third in in which case let's see okay so we get then a reasonable I mean this this Delta over our is then Delta to the two-thirds if Delta is small that's small so we're getting you know these parameters are small and so we're getting something that looks usefully like the sphere what do these theorems say here Delta over R cubed is on the order of magnitude of 1 and so we're saying that our manifold has bounded sectional curvatures that's what we want that's what the sphere does and the injectivity radius is at least R over 2 oops R is of the order of Delta to the one-third that's going to 0 so the weak point in the theorem is wait a minute we got lousy lower bounds for the injectivity radius what to do about that well we have the the following theorem if there is a manifold for which the injectivity radius is not so small our algorithm will find it and in fact our algorithm I mean the manifold that that we produce will be in this sense almost isometric to to that manifold ok so if there is a solution with reasonable injectivity radius we will find it but we don't know how to decide just by looking at the metric space whether there is a amount of fold with a reasonable injectivity radius ok well so that's that is the statement of the intrinsic version of the theorem so we've now covered I guess this is the usual the usual situation of for 4 speakers we've now covered what I thought would be the introduction to the talk so I have let's see how much time do I have left maybe 0 maybe 5 minutes what do you think right right all right let me not intrude on the break I will say a little bit about about the proofs for five minutes and I will I will I will quit that have passed faithfully okay all right let's let's look at the let's look at the intrinsic version of the problem so for the interent I'm sorry let's look at the extrinsic I think given that I have five minutes I should pick one or the other and I will pick the extrinsic version of the problem okay so let's alright if the reach is of the order of Tao let's let's take a small constant a small constant maybe depending on D times the reach and let's call that Delta zero and we're going to look on the length scale of Delta zero okay let's say let's say the Tao is one okay so this is a small constant or all right so I'm going to call this Delta zero now the manifold the manifold has reached at least one so it's curvatures are controlled and so if this is if this straight lines represents the tangent space then the manifold does something like that this is this distance is of the order of del zero but this distance is of the order of del zero squared okay so let's take let's take a projection PI an orthogonal projection on to this space which we hope is the tangent space to the manifold at this point and let's look at the distance the Euclidian distance between X and PI X and let's square it that's a more or less the distance squared to the manifold well maybe not if you're right on the manifold then then that's not a very good approximation so let's put here Delta 0 to the fourth Delta 0 squared quantity squared this is a pretty reliable estimation of the distance from X to the manifold squared plus Delta 0 to the fourth they're comparable you know the most a factor of 10 ok now that now imagine imagine we cover our manifold by not so many patches we have a compact manifold we've assumed that it's volume is not so big that ok we cover it by not so many patches and we put these functions together by a partition of unity ok then in a tubular neighborhood of where the manifold is we don't know where the manifold is we don't know whether there's a manifold but we've got ourselves a tubular neighborhood perhaps perhaps some of these balls are neighborhoods of the manifold and we don't actually see any points sampled from there because maybe our manifold is the sphere and all points come from the lower hemisphere so we may have to guess where these balls are and we exhaust over over a large set of guesses which is why our algorithm is so absurdly complicated and I mean computationally intensive but if we have guessed right then we have made ourselves a tubular neighborhood of the manifold and a function which looks an awful lot like the distance squared to the manifold well not quite the distance squared to the manifold the distance squared to the manifold plus some small constant so the Hessian of that locally has has lots of big if I ghen values of the order of magnitude of one and a few namely end the dimension of the manifold little eigenvalues close to zero it is then easy to look at something analogous to the critical points I don't I mean in the two minutes I don't have time to write down a precise definition but what one I mean there's a natural notion of a trough for such a for such a function you can make yourself an N dimensional manifold it is not close enough ok so the picture the picture that that arises is that the picture that arises is that here here is here is the true manifold and here is the manifold you construct okay and they are sort of close but not close enough however the manifold you construct has a normal bundle and there these manifolds are close enough that the manifold you want is a section of the normal bundle over the over this guy which we call the putative manifold and a section of modest norm so the problem is given given all right given the putative manifold and it's normal bundle and given a whole lot of data how do we use the data to construct the actual manifold that we want well because we've got a bundle over a known manifold we can reduce this to a local problem and the so the local problem is that we have some vector valued function on a ball in RN and we want to fit that to data and and and Charlie has been thinking about such questions for the last 15 years or so and so that's actually an easy case of the problems i've been thinking about for 15 years or so so do that and then that locally reconstructs this manifold as a section over this bundle you put those together with a partition of unity and you were done and I'm done with one minute to go thank you [Applause]

Channel: Institute for Advanced Study

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Id: wtmzRyrv9tQ

Length: 57min 38sec (3458 seconds)

Published: Sat Apr 07 2018