Betz limit for sailboats?

In the study of sustainable energy, there is a nice result on windmills, called the Betz limit: There is a finite limit to the fraction of the kinetic energy of the wind that a windmill can absorb or exploit. The reason is often stated as: If the windmill took all of the power in the wind, the wind would stop, and then there would be no flow of energy over the windmill. I'm not sure I exactly agree with that explanation, but let's leave that here.

On my travel home today I worked on the possibility that there is an equivalent to the Betz limit for sailboats. Is there an energetic way of looking at sailing that is useful?

One paradox is that a sailboat is sailing steadily when the net force on the boat is zero (just like when a windmill is turning at constant angular velocity). In the Betz limit, the windmill is thought of as having two different torques on it, one from the wind, and one from the turbine. Sailing has no turbine. So this problem has a conceptual component to it.

philosophy

I've been working on two philosophical projects this month. The first has been an interaction with Jim Peebles (Princeton) around a paper he has been writing, setting down his philosophy of physics. I am pretty aligned with his position, which I expect to hit the arXiv soon. I'm not a co-author of that. But one of the interesting things about science is how much of our work in in anonymous (or quasi-anonymous) support of others.

The second philosophical project is a paper about machine learning and science: I am trying to set down my thoughts about how ML can and can't help the sciences. This is fundamentally a philosophy-of-science question, not a science question.

radical papers I want to write (or will never write)

I have to finish my NSF proposal with Mike Blanton (NYU), so naturally I am in procrastination mode. Here are three papers I wish I would write. Maybe I should post them on my ideas blog:

Occam's Razor is wrong: This paper, co-authored with Jennifer Hill (NYU), would be about the fact that, in the real, observed world, the simplest explanation is always wrong or at least incomplete.

Causation is just causality: This paper, maybe co-authored with David Blei (Columbia) or Bernhard Schölkopf (MPI-IS) or Hill, shows that you don't need to have free will in order to have cogent causal explanations of data. That is, you don't need to phrase causality in terms of predictions for counter-factual experiments that you might have chosen to do.

You don't ever want evidence: This paper shows that any time you are computing the Bayesian evidence—what I call the fully marginalized likelihood (fml)—you are doing the wrong integral and solving the wrong problem. For both practical and theoretical (principled) reasons.

planning your science

I had two interactions today that made me think seriously about big-picture and design things. I like design language: How do you design your whole research program, and how do you design individual projects so they fit into it. One interaction was in the Astronomical Data Meeting at Flatiron, where Vivi Acquaviva (CUNY) talked about the intersection between what you are good at, what is important, and what brings you joy. That's a hard intersection to find. Or way too easy; I am not sure. The other interaction was a conversation with Jiayin Dong (Flatiron), who is thinking about faculty job applications and the like. How to talk about your research in terms of the next decade instead of the next year?

One comment that is frequently made by Hans-Walter Rix (MPIA) is that he feels like most early-career (and even mid-career) people spend too much time doing their science and not enough time planning and justifying their science. It is important to be able to answer “why” questions about your research, and in the medium term it helps all your projects.

gradients of unit vectors

When you work in a curvilinear coordinate system, and you need to take gradients or tensor derivatives of scalar, vector, and tensor functions, the gradients of the unit vectors appear in your expressions. The unit vectors have gradients because, in a curvilinear coordinate system, they have orientations that depend on position. I gestured and imagined and guessed these derivatives for a spherical coordinate system by thinking geometrically. I got strange expressions I didn't believe. Then, today, I checked them by painstakingly taking derivatives, and my intuitive derivatives turned out to be exactly correct?

spherical harmonics for tensor fields

I have been kicking around the generalization of spherical harmonics to vector spherical harmonics, and how that might generate the tensor spherical harmonics to all orders of tensor and all parities. I think I got it today! For every spherical harmonic (ell and em), there are three vector spherical harmonics obtained by multiplying by the radial vector, taking the transverse gradient, and taking the transverse gradient and crossing it into the radial direction. I think these can be generalized (using, say, the Ricci calculus) to make the 2-tensors and so on. If I am right, this is a new way to represent tensor fields on the sphere. Use cases: Cosmic backgrounds, and ocean dynamics.

patches of imaging

I am discussing with Sean Ku (NYU) and Victor Kuang (NYU) the NASA Cassini imaging of Saturn. We want to make (from the data) a high-quality face-on picture of the rings. This is a problem (from my perspective) in computer vision, so we need a camera model (and a lot of other things). One thing I hypothesized about this problem today is the following (am I right?):

Any sufficiently small patch of a camera image can be modeled with a pinhole-camera-like camera model, provided that we give the camera model the freedom to make the image plane not perpendicular to the line from the pinhole to the patch of the image plane. Is this correct? We are about to find out, the hard way.

what to say for Jim Peebles

I have been honored by an invitation to speak at a meeting in honor of Jim Peebles (Princeton) and his 2019 Nobel Prize in Physics. I spent the day working on things I want to say at this event. Obviously I have to be very critical of the Nobel Prize and all prizes! Haha. But I want to talk about large-scale structure and also the problem that we only ever get to observe one Universe. What does that mean for inductive reasoning and epistemology?

flexible models and interpolation

I gave the Königstuhl Colloquium today, which was fun. I spoke about this paper and related matters. I got great questions. One was about when to fit a very flexible model vs just doing simple interpolation. I gave some kind of minimal answer in my talk but then I thought about it a lot more later. The key difference between fitting a very flexible model and interpolation is that the former can be made part of a bigger probabilistic model whereas the latter (without serious modifications) cannot. That's a big deal when (say) you are trying to find planet transits in the face of stellar and spacecraft variability.

definition of the pseudoscalar, pseudovector, and pseudotensor

I am fully obsessed with geometry these days. In particular, I am obsessed with the point that scalars aren't just numbers, but rather numbers that don't depend on your choice of coordinate system. Similarly, vectors aren't just things with a magnitude and a direction: They are things with a magnitude and a direction which are coordinate free, or which have a stable direction and magnitude no matter what coordinate system you choose. Thus, for example, the unit vectors defining the x, y, and z directions of your coordinate system are not really vectors at all. But the acceleration due to gravity right here is a vector.

But there are pseudo- quantities. For example, the angular momentum isn't exactly a vector; it is a pseudo-vector: It's magnitude and direction doesn't depend on the orientation (or translation) of the coordinate system, but its direction does depend on the handedness of the coordinate system. Thus there are pseudoscalar, pseudovector, and pseudotensors in addition to scalars, vectors, and tensors. Today Soledad Villar (JHU) wrote definitions for these in the paper we are drafting. It isn't trivial, because we want a notation that is agnostic about the group operator and the thing it is operating on.

is machine learning good or bad for science?

I took the train to Vienna today, for a PhD defense and to give a talk about machine learning. My talk is interdisciplinary so I looked at how to generalize my arguments about astrophysics to all of the natural sciences. It turns out that this isn't as easy as I'd like, since it is hard to be specific outside of astrophysics! I'm going to learn a lot getting this talk ready.

coordinate freedom?

I spent the weekend recovering from the Gaia Fete. During my recovery day, I worked on very long-term projects. For example, I spent some time working on how to express the following issue in my work (with Villar) on exact symmetries:

The mathematics and computer-science communities call these exact symmetries “equivariances” and they are imagining that the data or the laws of physics are precisely equivariant in the sense that if you (say) rotate all the inputs, you get a rotated output. But this is not the main reason that we write the laws of physics in terms of exact symmetries! We write the laws of physics in terms of invariants because we want our laws of physics to be coordinate free. This is required even when the laws aren't equivariant! But I have trouble making this distinction clearly, since the mathematical implementations of the two symmetries are identical. There's some cool philosophy here: Does coordinate freedom enforce symmetries? What would it even look like for the laws of physics to be asymmetric but coordinate free?

epistemology and adversarial attacks

I am trying to prepare slides for a new talk (meaning: no recycled slides!) that I will be giving at MIT in the coming week. I am trying to boil down what's so troubling about machine learning in the natural sciences. I realized that what's so troubling (to me) is that the only standard of truth for a machine-learning model is how it performs on a finite set of held-out test data (usually drawn from the same distribution as the training data). That is nothing like the standard of truth for scientific models or theories! So, for example, a successful adversarial attack against a method is seen as a strange feature, when it comes to industrial machine-learning contexts. But I think that a successful adversarial attack completely invalidates a machine-learning method used in (many contexts in) natural science: It shows that the model isn't doing what we expect it to be doing. I hope to be able to articulate all this before Tuesday.

why does anyone need a bolometric correction?

As I mentioned yesterday, and against my better judgement, I am writing a long pedagogical document on magnitudes and how they work. I got to the bolometric-correction part and I realized that I didn't understand exactly why they are defined and how they are used. So I had a long think. The answer is: Theoretical models of stars are better at predicting the total bolometric luminosity output (energy per time) coming from a star than they are at predicting the detailed spectrum. Thus it makes sense to separate the belief about the total luminosity from the belief about the luminosity in any one band.

emulators

Right now, in cosmology, emulators are all the rage: Cosmological simulations are slow and expensive, and we need to speed them up! But I'm concerned: What if we use emulators in all our Euclid and LSST pipelines, and then we find something very surprising? What then? Do we check it all using full-blown simulations? But then, if so, why not do full-blown from the outset? Or do our standards of care depend on our results? They aren't supposed to, but maybe they do?

Worried about all this, I have been thinking about how to make emulators better, and give them provably better properties. It's why I've been working on building methods that exactly obey symmetries. But from an epistemological perspective, emulators might be the Worst Idea Ever^TM.

In detail, I'm working out how we might build emulators that are trained on one simulation suite but can be tested on another, made with a different cosmology and at a different mass resolution. I think work I've been doing with Villar (JHU) on the exact symmetries imposed by geometry and units might make this possible. I'm trying to figure out the scope of an achievable paper on this.

what are the assumptions underlying EPRV?

I reopened an old paper started (many years ago now) by Megan Bedell (Flatiron) and me, about the precision possible in extreme precision radial-velocity spectroscopy. Most of the results in the literature on how precisely you can measure a radial velocity (the information-theoretic question) depend on a very large number of assumptions, which we try to enumerate. The thing we'd like to do with this paper (and frankly, it will take many more after it) is to weaken or break those assumptions and see what gives. I have an intuition that if we understand all of that information theory, it will help us with observation planning and data analysis.

my own special FMM algorithm

I was in a location with no internet and no computing, so I spent an hour or so writing down what I think is the fast multipole method. It involves building an octtree, balanced in volume (not necessarily point content), and computing recursively the multipole amplitudes in all nodes (starting with the points in the leaves). Once that is done, at evaluation time, you do different things at different levels, depending on the radius out to which things are computed exactly. One thing I'm interested in is: Can you simplify that evaluation if you, say, build multiple trees?

representing spectrograph calibration images

In a spectrograph, many of the the raw calibration images are taken with something like arcs or flat-field lamps illuminating the slits or fibers, so that the spectral traces are lit up with calibration light. If the spectrograph flexes or changes, these traces move in two dimensions on the spectrograph image plane, or relative to the detector pixels. I am looking at whether we could have a generative model for these calibration images, so I spent the weekend toying with ways to represent the relevant kinds of small distortions of images. If the distortions are extremely small (as they are, say, for EXPRES) they can be represented with just a compact linear basis. But if they are a few pixels (as they appear to be for BOSS), I need a better representation.

the baryon-induced displacement field

Today Kate Storey-Fisher (NYU) showed me very nice visualizations of two matched simulations, one dark matter only, and one dark matter plus baryons. The simulations are matched in the sense that they have identical initial conditions, the only difference is that the latter simulation has baryon physics, such as cooling, star formation (approximately), AGN feedback (approximately), and so on. The simulations are from the IllustrisTNG project.

The thing that is interesting to us is whether we can model the differences between the simulations, and in particular whether building such a model will lead to insights about the fundamental physical mechanisms that lead to the differences. Large-scale gravity, after all, doesn't care about the small-scale composition or state of the matter; it only cares about the mass, so why are these simulations different at all? Of course there are lots of things about baryon physics that move matter, so it isn't a paradox, it's just interestingly non-trivial.

Now the question is: If we throw some gauge-invariant machine learning at this problem, will it lead to new insights about physical cosmology? That would be a real win.

coarse-graining a point cloud with a kd-tree?

As my loyal reader knows, I am interested in fast-multipole method and whether it could be used to improve or speed machine-learning methods on graphs or spatial point clouds. Over the last months, I have learned about lots of limitations of FMMs, some of which we discuss here. I'm still interested! But when I last spoke with Leslie Greengard (Flatiron) he indicated that he felt like if you want to take FMMs scale up to very clustered data in high dimensions, maybe you have to think of truly adaptive trees (not the fixed tree of an FMM), like perhaps kd-trees. Today Soledad villar (JHU) and I discussed this idea. The question is: What could be proved about such an approach, or are there such approaches where you could get things like accuracy guarantees? The FMM has the beautiful property that you can compute the precision of your approximation, and dial up the order to get better precision.