Nosimpler

Robust wireless power transfer using a nonlinear parity–time-symmetric circuit

Nature 546, 7658 (2017). doi:10.1038/nature22404

Authors: Sid Assawaworrarit, Xiaofang Yu & Shanhui Fan

Considerable progress in wireless power transfer has been made in the realm of non-radiative transfer, which employs magnetic-field coupling in the near field. A combination of circuit resonance and impedance transformation is often used to help to achieve efficient transfer of power over a predetermined distance of about the size of the resonators. The development of non-radiative wireless power transfer has paved the way towards real-world applications such as wireless powering of implantable medical devices and wireless charging of stationary electric vehicles. However, it remains a fundamental challenge to create a wireless power transfer system in which the transfer efficiency is robust against the variation of operating conditions. Here we propose theoretically and demonstrate experimentally that a parity–time-symmetric circuit incorporating a nonlinear gain saturation element provides robust wireless power transfer. Our results show that the transfer efficiency remains near unity over a distance variation of approximately one metre, without the need for any tuning. This is in contrast with conventional methods where high transfer efficiency can only be maintained by constantly tuning the frequency or the internal coupling parameters as the transfer distance or the relative orientation of the source and receiver units is varied. The use of a nonlinear parity–time-symmetric circuit should enable robust wireless power transfer to moving devices or vehicles.

Implantation of electrical probes into the brain has been central to both neuroscience research and biomedical applications, although conventional probes induce gliosis in surrounding tissue. We recently reported ultraflexible open mesh electronics implanted into rodent brains by syringe injection that exhibit promising chronic tissue response and recording stability. Here we...

Author(s): Adhip Agarwala and Vijay B. Shenoy

Topological insulators, so far only identified in materials with an ordered crystal structure, could potentially be found in amorphous materials.

[Phys. Rev. Lett. 118, 236402] Published Thu Jun 08, 2017

Collective motion in biology is often modelled as a dynamical system, in which individuals are represented as particles whose interactions are determined by the current state of the system. Many animals, however, including humans, have predictive capabilities, and presumably base their behavioural decisions---at least partially---upon an anticipated state of their environment. We explore a minimal version of this idea in the context of particles that interact according to a pairwise potential. Anticipation enters the picture by calculating the interparticle forces from linear extrapolations of the particle positions some time $\tau$ into the future. Simulations show that for intermediate values of $\tau$, compared to a transient time scale defined by the potential and the initial conditions, the particles form rotating clusters in which the particles are arranged in a hexagonal pattern. Analysis of the system shows that anticipation induces energy dissipation and we show that the kinetic energy asymptotically decays as $1/t$. Furthermore, we show that the angular momentum is not necessarily conserved for $\tau >0$, and that asymmetries in the initial condition therefore can cause rotational movement. These results suggest that anticipation could play an important role in collective behaviour, since it induces pattern formation and stabilises the dynamics of the system.

Ring attractors are a class of recurrent networks hypothesized to underlie the representation of heading direction. Such network structures, schematized as a ring of neurons whose connectivity depends on their heading preferences, can sustain a bump-like activity pattern whose location can be updated by continuous shifts along either turn direction. We recently reported that a population of fly neurons represents the animal’s heading via bump-like activity dynamics. We combined two-photon calcium imaging in head-fixed flying flies with optogenetics to overwrite the existing population representation with an artificial one, which was then maintained by the circuit with naturalistic dynamics. A network with local excitation and global inhibition enforces this unique and persistent heading representation. Ring attractor networks have long been invoked in theoretical work; our study provides physiological evidence of their existence and functional architecture.

Nature Neuroscience 20, 886 (2017). doi:10.1038/nn.4548

Authors: Zhisong He, Dingding Han, Olga Efimova, Patricia Guijarro, Qianhui Yu, Anna Oleksiak, Shasha Jiang, Konstantin Anokhin, Boris Velichkovsky, Stefan Grünewald & Philipp Khaitovich

Expressive eye widening (as in fear) and eye narrowing (as in disgust) are associated with opposing optical consequences and serve opposing perceptual functions. Lee and Anderson suggest that the opposing effects of eye widening and narrowing on the expresser’s visual perception have been socially co-opted to denote opposing mental states of sensitivity and discrimination, respectively, such that opposing complex mental states may originate from this simple perceptual opposition. Their abstract:

Human eyes convey a remarkable variety of complex social and emotional information. However, it is unknown which physical eye features convey mental states and how that came about. In the current experiments, we tested the hypothesis that the receiver’s perception of mental states is grounded in expressive eye appearance that serves an optical function for the sender. Specifically, opposing features of eye widening versus eye narrowing that regulate sensitivity versus discrimination not only conveyed their associated basic emotions (e.g., fear vs. disgust, respectively) but also conveyed opposing clusters of complex mental states that communicate sensitivity versus discrimination (e.g., awe vs. suspicion). This sensitivity-discrimination dimension accounted for the majority of variance in perceived mental states (61.7%). Further, these eye features remained diagnostic of these complex mental states even in the context of competing information from the lower face. These results demonstrate that how humans read complex mental states may be derived from a basic optical principle of how people see.

Here’s a video of the talk I gave at the Stanford Complexity Group:

You can see slides here:

• Biology as information dynamics.

Abstract. If biology is the study of self-replicating entities, and we want to understand the role of information, it makes sense to see how information theory is connected to the ‘replicator equation’ — a simple model of population dynamics for self-replicating entities. The relevant concept of information turns out to be the information of one probability distribution relative to another, also known as the Kullback–Liebler divergence. Using this we can get a new outlook on free energy, see evolution as a learning process, and give a clearer, more general formulation of Fisher’s fundamental theorem of natural selection.

I’d given a version of this talk earlier this year at a workshop on Quantifying biological complexity, but I’m glad this second try got videotaped and not the first, because I was a lot happier about my talk this time. And as you’ll see at the end, there were a lot of interesting questions.

Excerpts from an interesting article in this month's The Atlantic:

Our ability to know the price of anything, anytime, anywhere, has given us, the consumers, so much power that retailers—in a desperate effort to regain the upper hand, or at least avoid extinction—are now staring back through the screen. They are comparison shopping us...

The price of a can of soda in a vending machine can now vary with the temperature outside. The price of the headphones Google recommends may depend on how budget-conscious your web history shows you to be, one study found. For shoppers, that means price—not the one offered to you right now, but the one offered to you 20 minutes from now, or the one offered to me, or to your neighbor—may become an increasingly unknowable thing...

Four researchers in Catalonia tried to answer the question with dummy computers that mimicked the web-browsing patterns of either “affluent” or “budget conscious” customers for a week. When the personae went “shopping,” they weren’t shown different prices for the same goods. They were shown different goods. The average price of the headphones suggested for the affluent personae was four times the price of those suggested for the budget-conscious personae. Another experiment demonstrated a more direct form of price discrimination: Computers with addresses in greater Boston were shown lower prices than those in more-remote parts of Massachusetts on identical goods...

More at the link.

[cat picture]

Kevin Lewis pointed me to this article:

There are several methods for building hype. The wealth of currently available public relations techniques usually forces the promoter to judge, a priori, what will likely be the best method. Meta-hype is a methodology that facilitates this decision by combining all identified hype algorithms pertinent for a particular promotion problem. Meta-hype generates a final press release that is at least as good as any of the other models considered for hyping the claim. The overarching aim of this work is to introduce meta-hype to analysts and practitioners. This work compares the performance of journal publication, preprints, blogs, twitter, Ted talks, NPR, and meta-hype to predict successful promotion. A nationwide database including 89,013 articles, tweets, and news stories. All algorithms were evaluated using the total publicity value (TPV) in a test sample that was not included in the training sample used to fit the prediction models. TPV for the models ranged between 0.693 and 0.720. Meta-hype was superior to all but one of the algorithms compared. An explanation of meta-hype steps is provided. Meta-hype is the first step in targeted hype, an analytic framework that yields double hyped promotion with fewer assumptions than the usual publicity methods. Different aspects of meta-hype depending on the context, its function within the targeted promotion framework, and the benefits of this methodology in the addiction to extreme claims are discussed.

I can’t seem to find the link right now, but you get the idea.

P.S. The above is a parody of the abstract of a recent article on “super learning” by Acion et al. I did not include a link because the parody was not supposed to be a criticism of the content of the paper in any way; I just thought some of the wording in the abstract was kinda funny. Indeed, I thought I’d disguised the abstract enough that no one would make the connection but I guess Google is more powerful than I’d realized.

But this discussion by Erin in the comment thread revealed that some people were taking this post in a way I had not intended. So I added this comment.

tl;dr: I’m not criticizing the content of the Acion et al. paper in any way, and the above post was not intended to be seen as such a criticism.

The post The meta-hype algorithm appeared first on Statistical Modeling, Causal Inference, and Social Science.

Journal of Cognitive Neuroscience, Volume 29, Issue 10, Page 1684-1698, October 2017.

The post Las Vegas average is over no arbitrage condition appeared first on Marginal REVOLUTION.

Are emergent properties really for losers? Why are architectures important? What are “mirror neuron ensembles” anyway? My last post presented an idiosyncratic distillation of the Big Ideas in Cognitive Neuroscience symposium, presented by six speakers at the 2017 CNS meeting. Here I’ll briefly explain what I meant in the bullet points. In some cases I didn't quite understand what the speaker meant so I used outside sources. At the end is a bonus reading list.

The first two speakers made an especially fun pair on the topic of memory: they held opposing views on the “engram”, the physical manifestation of a memory in the brain.¹ They also disagreed on most everything else.


1. Charles Randy Gallistel (Rutgers University) – What Memory Must Look Like

Gallistel is convinced that Most Neuroscientists Are Wrong About the Brain. This subtly bizarre essay in Nautilus (which was widely scorned on Twitter) succinctly summarized the major points of his talk. You and I may think the brain-as-computer metaphor has outlived its usefulness, but Gallistel says that “Computation in the brain must resemble computation in a computer.” 

• The brain is an information processing device in the sense of Shannon information theory.

Shannon information is a set of possible messages encoded as bit patterns and sent over a noisy channel to a recipient that will hopefully decode the message with minimal error. In this purely mathematical theory, the semantic content (meaning) of a message is irrelevant. The brain stores numbers and that's that.

• Memories (“engrams”) are not stored at synapses.

Instead, engrams reside in molecules inside cells. The brain “encodes information into molecules inside neurons and reads out that information for use in computational operations.” A 2014 paper on conditioned responses in cerebellar Purkinje cells was instrumental in overturning synaptic plasticity (strengthening or weakening of synaptic connections) as the central mechanism for learning and memory, according to Gallistel.² Most other scientists do not share this view.³

• The engram is inter-spike interval.

Spike train solutions based on rate coding are wrong. Meaning, the code is not conveyed by the firing rate of neurons. Instead, numbers are conveyed to engrams via a combinatorial interspike interval code. Engrams then reside in cell-intrinsic molecular structures. In the end, memory must look like the DNA code.

• Emergent properties are for losers.

“Emergent property” is a code word for “we don't know.”



2. Tomás Ryan (@TJRyan_77) – Information Storage in Memory Engrams

Ryan began by acknowledging that he had tremendous respect for Gallistal's speech — which was in turn powerful, illuminating, very categorical, polarizing, and rigid. But wrong. Oh so very wrong. Memory is not essentially molecular, we should not approach memory and the brain from a design perspective, and information storage need not mimic a computer.

• The brain does not use Shannon information.

More precisely, “the kind of information the brain uses may be very different from Shannon information.” Why is that? Brains evolved, in kludgy ways that don't resemble a computer. The information used by the brain may be encoded without having to reduce it to Shannon form, and may not be quantifiable as units.

• Memories (“engrams”) are not stored at synapses.

Memory is not stored by changes in synaptic weights, Ryan and Gallistel agree on this. The dominant view has been falsified by a number of studies — including one by Ryan and colleagues that used engram labeling. Specific “engram cells” can be labeled during learning using optogenetic techniques, and later stimulated to induce the recall of specific memories. These memories can be reactivated even after protein synthesis inhibitors have (1) induced amnesia, and (2) prevented the usual memory consolidation-related changes in synaptic strength.

• We learn entirely through spike trains.

Spike trains are necessary but not sufficient to explain how information is coded in the brain. On the other hand, instincts are transmitted genetically and are not learned via spike trains.

• The engram is an emergent property.

And fitting with all of the above, “the engram is an emergent property mediated through synaptic connections” (not through synaptic weights). Stable connectivity is what stores information, not molecules.


Angela Friederici (Max Planck Institute for Human Cognitive and Brain Sciences) – Structure and Dynamics of the Language Network

Following on the heels of the rodent engram crowd, Friederici pointed out the obvious limitations of studying language as a human trait.

• Language is genetically predetermined.

The human ability to acquire language is based on a genetically predetermined structural neural network. Although the degree of innateness has been disputed, a bias or propensity of brain development towards particular modes of information processing is less controversial. According to Friederici, language capacity is rooted in “merge”, a specific computation that binds words together to form phrases and sentences.

• The “merge” computation is localized in BA 44.

This wasn't one of my original bullet points, but I found this statement rather surprising and unbelievable. It implies that our capacity for language is located in the anterior ventral portion of Brodmann's area 44 in the left hemisphere (the tiny red area in the PHRASE > LIST panel below).



The problem is that acute stroke patients with dysfunctional tissue in left BA 44 do not have impaired syntax. Instead, they have difficulty with phonological short-term memory (keeping strings of digits in mind, like remembering a phone number).

• There is something called mirror neural ensembles.

I'll just have to leave this slide here, since I really didn't understand it, even on the second viewing.



“This is a poor hypothesis,” she said.


Jean-Rémi King (@jrking0) – Parsing Human Minds

King's expertise is in visual processing (not language), but his talk drew parallels between vision and speech comprehension. A key goal in both domains is to identify the algorithm (sequence of operations) that translates input into meaning.

• Recursion is big. 

Despite these commonalities, the structure of language presents the unique challenge of nesting (or recursion): each constituent in a sentence can be made of subconstituents of the same nature, which can result in ambiguity.

• Architectures are important. 

Decoding aspects of a sensory stimulus using MEG and machine learning is lovely, but it doesn't tell you the algorithm. What is the computational architecture? Is it sustained or feedforward or recurrent?

Each architecture could be compatible with a pattern of brain activity at different time points. But do the classifiers at different time points generalize to other time points? This can be determined by a temporal generalization analysis, which “reveals a repertoire of canonical brain dynamics.”


Danielle Bassett (@DaniSBassett) – A Network Neuroscience of Human Learning: Potential to Inform Quantitative Theories of Brain and Behavior

Bassett previewed an arc of exciting ideas where we've shown progress, followed by frustrations and failures, which may ultimately provide an opening for the really Big Ideas. Her focus is on learning from a network perspective, which means patterns of connectivity in the whole brain. What is the underlying network architecture that facilitates the spatial distributed effects?



What is the relationship between these two notions of modularity?
[I ask this as an honest question.]

Major challenges remain, of course.

• Build a bridge from networks to models of behavior.

• Use generative models to construct theories. 

John Krakauer (@blamlab) – Big Ideas in Cognitive Neuroscience: Action

Krakauer mentioned the Big Questions in Neuroscience symposium at the 2016 SFN meeting, which motivated the CNS symposium as well as a splashy critical paper in Neuron. He raised an interesting point about how the term “connectivity” has different meanings, i.e. the type of embedded connectivity that stores information (engrams) vs. the type of correlational connectivity when modules combine with each other to produce behavior. [BTW, is everyone here using “modules” in the same way?]

• Machine learning will save us. 

Krakauer discussed work on motor learning using adaptation paradigms and simple execution tasks. But there's a dirty secret: there is no computational model, no algorithmic theory of how practice makes you better on those tasks. Can the computational view get an upgrade from machine learning? Go out and read the manifesto by Marblestone, Wayne, and Kording: Toward an Integration of Deep Learning and Neuroscience. And you better learn about cost functions, because they're very important.⁴

• Go back to behavioral neuroscience.

This is the only way to work out the right cost functions. Bottom line: Networks represent weighting modules into the cost function.⁴ 

Here's Why Most Neuroscientists Are Wrong About the Brain. Gallistel in Nautilus, Oct. 2015.

Time to rethink the neural mechanisms of learning and memory. Gallistel CR, Balsam PD. Neurobiol Learn Mem. 2014 Feb;108:136-44.

Engrams are Cool

What is memory? The present state of the engram. Poo MM, Pignatelli M, Ryan TJ, Tonegawa S, Bonhoeffer T, Martin KC, Rudenko A, Tsai LH, Tsien RW, Fishell G, Mullins C, Gonçalves JT, Shtrahman M, Johnston ST,  Gage FH, Dan Y, Long J, Buzsáki G, Stevens C. BMC Biol. 2016 May 19;14:40.

Engram cells retain memory under retrograde amnesia. Ryan TJ, Roy DS, Pignatelli M, Arons A, Tonegawa S. Science. 2015 May 29;348(6238):1007-13.

Engrams are Overrated

For good measure, some contrarian thoughts floating around Twitter...


“Can We Localize Merge in the Brain? Yes We Can”

Merge in the Human Brain: A Sub-Region Based Functional Investigation in the Left Pars Opercularis. Zaccarella E, Friederici AD. Front Psychol. 2015 Nov 27;6:1818.

The neurobiological nature of syntactic hierarchies. Zaccarella E, Friederici AD. Neurosci Biobehav Rev. 2016 Jul 29. doi: 10.1016/j.neubiorev.2016.07.038.

Really?

Asyntactic comprehension, working memory, and acute ischemia in Broca's area versus angular gyrus. Newhart M, Trupe LA, Gomez Y, Cloutman L, Molitoris JJ, Davis C, Leigh R, Gottesman RF, Race D, Hillis AE.  Cortex. 2012 Nov-Dec;48(10):1288-97.

Patients with acute strokes in left BA 44 (part of Broca's area) do not have impaired syntax.


Dynamics of Mental Representations

Characterizing the dynamics of mental representations: the temporal generalization method. King JR, Dehaene S. Trends Cogn Sci. 2014 Apr;18(4):203-10.

King JR, Pescetelli N, Dehaene S. Brain Mechanisms Underlying the Brief Maintenance of Seen and Unseen Sensory Information. Neuron. 2016; 92(5):1122-1134.


A Spate of New Network Articles by Bassett

A Network Neuroscience of Human Learning: Potential to Inform Quantitative Theories of Brain and Behavior. Bassett DS, Mattar MG. Trends Cogn Sci. 2017 Apr;21(4):250-264.

This one is most relevant to Dr. Bassett's talk, as it is the title of her talk.

Network neuroscience. Bassett DS, Sporns O. Nat Neurosci. 2017 Feb 23;20(3):353-364.

Emerging Frontiers of Neuroengineering: A Network Science of Brain Connectivity. Bassett DS, Khambhati AN, Grafton ST. Annu Rev Biomed Eng. 2017 Mar 27. doi: 10.1146/annurev-bioeng-071516-044511.

Modelling And Interpreting Network Dynamics [bioRxiv preprint]. Ankit N Khambhati, Ann E Sizemore, Richard F Betzel, Danielle S Bassett. doi: https://doi.org/10.1101/124016


Behavior is Underrated

Neuroscience Needs Behavior: Correcting a Reductionist Bias. Krakauer JW, Ghazanfar AA, Gomez-Marin A, MacIver MA, Poeppel D. Neuron. 2017 Feb 8;93(3):480-490.

The first author was a presenter and the last author an organizer of the symposium.

"Before I knew anything about language acquisition, I assumed that babies making these utterances were referring to their parents. But this interpretation is backwards: mama/papa words just happen to be the easiest word-like sounds for babies to make.  The sounds came first – as experiments in vocalization – and parents adopted them as pet names for themselves."

Presented at Sentence First in a post introducing a new crowdsourced language project (details at the links).

How many groups of order four are there? Technically, there are an enormous number, so much so, in fact, that the class of groups of order four is not even a set, but merely a proper class. This is because any four objects can be turned into a group by designating one of the four objects, say , to be the group identity, and imposing a suitable multiplication table (and inversion law) on the four elements in a manner that obeys the usual group axioms. Since all sets are themselves objects, the class of four-element groups is thus at least as large as the class of all sets, which by Russell’s paradox is known not to itself be a set (assuming the usual ZFC axioms of set theory).

A much better question is to ask how many groups of order four there are up to isomorphism, counting each isomorphism class of groups exactly once. Now, as one learns in undergraduate group theory classes, the answer is just “two”: the cyclic group and the Klein four-group .

More generally, given a class of objects and some equivalence relation on (which one should interpret as describing the property of two objects in being “isomorphic”), one can consider the number of objects in “up to isomorphism”, which is simply the cardinality of the collection of equivalence classes of . In the case where is finite, one can express this cardinality by the formula

thus one counts elements in , weighted by the reciprocal of the number of objects they are isomorphic to.

Example 1 Let be the five-element set of integers between and . Let us say that two elements of are isomorphic if they have the same magnitude: . Then the quotient space consists of just three equivalence classes: , , and . Thus there are three objects in up to isomorphism, and the identity (1) is then just

Thus, to count elements in up to equivalence, the elements are given a weight of because they are each isomorphic to two elements in , while the element is given a weight of because it is isomorphic to just one element in (namely, itself).

Given a finite probability set , there is also a natural probability distribution on , namely the uniform distribution, according to which a random variable is set equal to any given element of with probability :

Given a notion of isomorphism on , one can then define the random equivalence class that the random element belongs to. But if the isomorphism classes are unequal in size, we now encounter a biasing effect: even if was drawn uniformly from , the equivalence class need not be uniformly distributed in . For instance, in the above example, if was drawn uniformly from , then the equivalence class will not be uniformly distributed in the three-element space , because it has a probability of being equal to the class or to the class , and only a probability of being equal to the class .

However, it is possible to remove this bias by changing the weighting in (1), and thus changing the notion of what cardinality means. To do this, we generalise the previous situation. Instead of considering sets with an equivalence relation to capture the notion of isomorphism, we instead consider groupoids, which are sets in which there are allowed to be multiple isomorphisms between elements in (and in particular, there are allowed to be multiple automorphisms from an element to itself). More precisely:

Definition 2 A groupoid is a set (or proper class) , together with a (possibly empty) collection of “isomorphisms” between any pair of elements of , and a composition map from isomorphisms , to isomorphisms in for every , obeying the following group-like axioms:

• (Identity) For every , there is an identity isomorphism , such that for all and .
• (Associativity) If , , and for some , then .
• (Inverse) If for some , then there exists an inverse isomorphism such that and .

We say that two elements of a groupoid are isomorphic, and write , if there is at least one isomorphism from to .

Example 3 Any category gives a groupoid by taking to be the set (or class) of objects, and to be the collection of invertible morphisms from to . For instance, in the category of sets, would be the collection of bijections from to ; in the category of linear vector spaces over some given base field , would be the collection of invertible linear transformations from to ; and so forth.

Every set equipped with an equivalence relation can be turned into a groupoid by assigning precisely one isomorphism from to for any pair with , and no isomorphisms from to when , with the groupoid operations of identity, composition, and inverse defined in the only way possible consistent with the axioms. We will call this the simply connected groupoid associated with this equivalence relation. For instance, with as above, if we turn into a simply connected groupoid, there will be precisely one isomorphism from to , and also precisely one isomorphism from to , but no isomorphisms from to , , or .

However, one can also form multiply-connected groupoids in which there can be multiple isomorphisms from one element of to another. For instance, one can view as a space that is acted on by multiplication by the two-element group . This gives rise to two types of isomorphisms, an identity isomorphism from to for each , and a negation isomorphism from to for each ; in particular, there are two automorphisms of (i.e., isomorphisms from to itself), namely and , whereas the other four elements of only have a single automorphism (the identity isomorphism). One defines composition, identity, and inverse in this groupoid in the obvious fashion (using the group law of the two-element group ); for instance, we have .

For a finite multiply-connected groupoid, it turns out that the natural notion of “cardinality” (or as I prefer to call it, “cardinality up to isomorphism”) is given by the variant

of (1). That is to say, in the multiply connected case, the denominator is no longer the number of objects isomorphic to , but rather the number of isomorphisms from to other objects . Grouping together all summands coming from a single equivalence class in , we can also write this expression as

where is the automorphism group of , that is to say the group of isomorphisms from to itself. (Note that if belong to the same equivalence class , then the two groups and will be isomorphic and thus have the same cardinality, and so the above expression is well-defined.

For instance, if we take to be the simply connected groupoid on , then the number of elements of up to isomorphism is

exactly as before. If however we take the multiply connected groupoid on , in which has two automorphisms, the number of elements of up to isomorphism is now the smaller quantity

the equivalence class is now counted with weight rather than due to the two automorphisms on . Geometrically, one can think of this groupoid as being formed by taking the five-element set , and “folding it in half” around the fixed point , giving rise to two “full” quotient points and one “half” point . More generally, given a finite group acting on a finite set , and forming the associated multiply connected groupoid, the cardinality up to isomorphism of this groupoid will be , since each element of will have isomorphisms on it (whether they be to the same element , or to other elements of ).

The definition (2) can also make sense for some infinite groupoids; to my knowledge this was first explicitly done in this paper of Baez and Dolan. Consider for instance the category of finite sets, with isomorphisms given by bijections as in Example 3. Every finite set is isomorphic to for some natural number , so the equivalence classes of may be indexed by the natural numbers. The automorphism group of has order , so the cardinality of up to isomorphism is

(This fact is sometimes loosely stated as “the number of finite sets is “, but I view this statement as somewhat misleading if the qualifier “up to isomorphism” is not added.) Similarly, when one allows for multiple isomorphisms from a group to itself, the number of groups of order four up to isomorphism is now

because the cyclic group has two automorphisms, whereas the Klein four-group has six.

In the case that the cardinality of a groupoid up to isomorphism is finite and non-empty, one can now define the notion of a random isomorphism class in drawn “uniformly up to isomorphism”, by requiring the probability of attaining any given isomorphism class to be

thus the probability of being isomorphic to a given element will be inversely proportional to the number of automorphisms that has. For instance, if we take to be the set with the simply connected groupoid, will be drawn uniformly from the three available equivalence classes , with a probability of attaining each; but if instead one uses the multiply connected groupoid coming from the action of , and draws uniformly up to isomorphism, then and will now be selected with probability each, and will be selected with probability . Thus this distribution has accounted for the bias mentioned previously: if a finite group acts on a finite space , and is drawn uniformly from , then now still be drawn uniformly up to isomorphism from , if we use the multiply connected groupoid coming from the action, rather than the simply connected groupoid coming from just the -orbit structure on .

Using the groupoid of finite sets, we see that a finite set chosen uniformly up to isomorphism will have a cardinality that is distributed according to the Poisson distribution of parameter , that is to say it will be of cardinality with probability .

One important source of groupoids are the fundamental groupoids of a manifold (one can also consider more general topological spaces than manifolds, but for simplicity we will restrict this discussion to the manifold case), in which the underlying space is simply , and the isomorphisms from to are the equivalence classes of paths from to up to homotopy; in particular, the automorphism group of any given point is just the fundamental group of at that base point. The equivalence class of a point in is then the connected component of in . The cardinality up to isomorphism of the fundamental groupoid is then

where is the collection of connected components of , and is the order of the fundamental group of . Thus, simply connected components of count for a full unit of cardinality, whereas multiply connected components (which can be viewed as quotients of their simply connected cover by their fundamental group) will count for a fractional unit of cardinality, inversely to the order of their fundamental group.

This notion of cardinality up to isomorphism of a groupoid behaves well with respect to various basic notions. For instance, suppose one has an -fold covering map of one finite groupoid by another . This means that is a functor that is surjective, with all preimages of cardinality , with the property that given any pair in the base space and any in the preimage of , every isomorphism has a unique lift from the given initial point (and some in the preimage of ). Then one can check that the cardinality up to isomorphism of is times the cardinality up to isomorphism of , which fits well with the geometric picture of as the -fold cover of . (For instance, if one covers a manifold with finite fundamental group by its universal cover, this is a -fold cover, the base has cardinality up to isomorphism, and the universal cover has cardinality one up to isomorphism.) Related to this, if one draws an equivalence class of uniformly up to isomorphism, then will be an equivalence class of drawn uniformly up to isomorphism also.

Indeed, one can show that this notion of cardinality up to isomorphism for groupoids is uniquely determined by a small number of axioms such as these (similar to the axioms that determine Euler characteristic); see this blog post of Qiaochu Yuan for details.

The probability distributions on isomorphism classes described by the above recipe seem to arise naturally in many applications. For instance, if one draws a profinite abelian group up to isomorphism at random in this fashion (so that each isomorphism class of a profinite abelian group occurs with probability inversely proportional to the number of automorphisms of this group), then the resulting distribution is known as the Cohen-Lenstra distribution, and seems to emerge as the natural asymptotic distribution of many randomly generated profinite abelian groups in number theory and combinatorics, such as the class groups of random quadratic fields; see this previous blog post for more discussion. For a simple combinatorial example, the set of fixed points of a random permutation on elements will have a cardinality that converges in distribution to the Poisson distribution of rate (as discussed in this previous post), thus we see that the fixed points of a large random permutation asymptotically are distributed uniformly up to isomorphism. I’ve been told that this notion of cardinality up to isomorphism is also particularly compatible with stacks (which are a good framework to describe such objects as moduli spaces of algebraic varieties up to isomorphism), though I am not sufficiently acquainted with this theory to say much more than this.

Filed under: expository, math.CO, math.GR, math.GT Tagged: groupoid cardinality

What is the value of the whole in terms of the values of the parts?

More specifically, given a finite set whose elements have assigned “values” v 1,…,v nv_1, \ldots, v_n and assigned “sizes” p 1,…,p np_1, \ldots, p_n (normalized to sum to 11), how can we assign a value σ(p,v)\sigma(\mathbf{p}, \mathbf{v}) to the set in a coherent way?

This seems like a very general question. But in fact, just a few sensible requirements on the function σ\sigma are enough to pin it down almost uniquely. And the answer turns out to be closely connected to existing mathematical concepts that you probably already know.

Let’s write

Δ n={(p 1,…,p n)∈ℝ n:p i≥0,∑p i=1} \Delta_n = \Bigl\{ (p_1, \ldots, p_n) \in \mathbb{R}^n : p_i \geq 0, \sum p_i = 1 \Bigr\}

for the set of probability distributions on {1,…,n}\{1, \ldots, n\}. Assuming that our “values” are positive real numbers, we’re interested in sequences of functions

(σ:Δ n×(0,∞) n→(0,∞)) n≥1 \Bigl( \sigma \colon \Delta_n \times (0, \infty)^n \to (0, \infty) \Bigr)_{n \geq 1}

that aggregate the values of the elements to give a value to the whole set. So, if the elements of the set have relative sizes p=(p 1,…,p n)\mathbf{p} = (p_1, \ldots, p_n) and values v=(v 1,…,v n)\mathbf{v} = (v_1, \ldots, v_n), then the value assigned to the whole set is σ(p,v)\sigma(\mathbf{p}, \mathbf{v}).

Here are some properties that it would be reasonable for σ\sigma to satisfy.

Homogeneity  The idea is that whatever “value” means, the value of the set and the value of the elements should be measured in the same units. For instance, if the elements are valued in kilograms then the set should be valued in kilograms too. A switch from kilograms to grams would then multiply both values by 1000. So, in general, we ask that

σ(p,cv)=cσ(p,v) \sigma(\mathbf{p}, c\mathbf{v}) = c \sigma(\mathbf{p}, \mathbf{v})

for all p∈Δ n\mathbf{p} \in \Delta_n, v∈(0,∞) n\mathbf{v} \in (0, \infty)^n and c∈(0,∞)c \in (0, \infty).

Monotonicity  The values of the elements are supposed to make a positive contribution to the value of the whole, so we ask that if v i≤v′ iv_i \leq v'_i for all ii then

σ(p,v)≤σ(p,v′) \sigma(\mathbf{p}, \mathbf{v}) \leq \sigma(\mathbf{p}, \mathbf{v}')

for all p∈Δ n\mathbf{p} \in \Delta_n.

Replication  Suppose that our nn elements have the same size and the same value, vv. Then the value of the whole set should be nvn v. This property says, among other things, that σ\sigma isn’t an average: putting in more elements of value vv increases the value of the whole set!

If σ\sigma is homogeneous, we might as well assume that v=1v = 1, in which case the requirement is that

σ((1/n,…,1/n),(1,…,1))=n. \sigma\bigl( (1/n, \ldots, 1/n), (1, \ldots, 1) \bigr) = n.

Modularity  This one’s a basic logical axiom, best illustrated by an example.

Imagine that we’re very ambitious and wish to evaluate the entire planet — or at least, the part that’s land. And suppose we already know the values and relative sizes of every country.

We could, of course, simply put this data into σ\sigma and get an answer immediately. But we could instead begin by evaluating each continent, and then compute the value of the planet using the values and sizes of the continents. If σ\sigma is sensible, this should give the same answer.

The notation needed to express this formally is a bit heavy. Let w∈Δ n\mathbf{w} \in \Delta_n; in our example, n=7n = 7 (or however many continents there are) and w=(w 1,…,w 7)\mathbf{w} = (w_1, \ldots, w_7) encodes their relative sizes. For each i=1,…,ni = 1, \ldots, n, let p i∈Δ k i\mathbf{p}^i \in \Delta_{k_i}; in our example, p i\mathbf{p}^i encodes the relative sizes of the countries on the iith continent. Then we get a probability distribution

w∘(p 1,…,p n)=(w 1p 1 1,…,w 1p k 1 1,…,w np 1 n,…,w np k n n)∈Δ k 1+⋯+k n, \mathbf{w} \circ (\mathbf{p}^1, \ldots, \mathbf{p}^n) = (w_1 p^1_1, \ldots, w_1 p^1_{k_1}, \,\,\ldots, \,\, w_n p^n_1, \ldots, w_n p^n_{k_n}) \in \Delta_{k_1 + \cdots + k_n},

which in our example encodes the relative sizes of all the countries on the planet. (Incidentally, this composition makes (Δ n)(\Delta_n) into an operad, a fact that we’ve discussed many times before on this blog.) Also let

v 1=(v 1 1,…,v k 1 1)∈(0,∞) k 1,…,v n=(v 1 n,…,v k n n)∈(0,∞) k n. \mathbf{v}^1 = (v^1_1, \ldots, v^1_{k_1}) \in (0, \infty)^{k_1}, \,\,\ldots,\,\, \mathbf{v}^n = (v^n_1, \ldots, v^n_{k_n}) \in (0, \infty)^{k_n}.

In the example, v j iv^i_j is the value of the jjth country on the iith continent. Then the value of the iith continent is σ(p i,v i)\sigma(\mathbf{p}^i, \mathbf{v}^i), so the axiom is that

σ(w∘(p 1,…,p n),(v 1 1,…,v k 1 1,…,v 1 n,…,v k n n))=σ(w,(σ(p 1,v 1),…,σ(p n,v n))). \sigma \bigl( \mathbf{w} \circ (\mathbf{p}^1, \ldots, \mathbf{p}^n), (v^1_1, \ldots, v^1_{k_1}, \ldots, v^n_1, \ldots, v^n_{k_n}) \bigr) = \sigma \Bigl( \mathbf{w}, \bigl( \sigma(\mathbf{p}^1, \mathbf{v}^1), \ldots, \sigma(\mathbf{p}^n, \mathbf{v}^n) \bigr) \Bigr).

The left-hand side is the value of the planet calculated in a single step, and the right-hand side is its value when calculated in two steps, with continents as the intermediate stage.

Symmetry  It shouldn’t matter what order we list the elements in. So it’s natural to ask that

σ(p,v)=σ(pτ,vτ) \sigma(\mathbf{p}, \mathbf{v}) = \sigma(\mathbf{p} \tau, \mathbf{v} \tau)

for any τ\tau in the symmetric group S nS_n, where the right-hand side refers to the obvious S nS_n-actions.

Absent elements should count for nothing! In other words, if p 1=0p_1 = 0 then we should have

σ((p 1,…,p n),(v 1,…,v n))=σ((p 2,…,p n),(v 2,…,v n)). \sigma\bigl( (p_1, \ldots, p_n), (v_1, \ldots, v_n)\bigr) = \sigma\bigl( (p_2, \ldots, p_n), (v_2, \ldots, v_n)\bigr).

This isn’t quite triival. I haven’t yet given you any examples of the kind of function that σ\sigma might be, but perhaps you already have in mind a simple one like this:

σ(p,v)=v 1+⋯+v n. \sigma(\mathbf{p}, \mathbf{v}) = v_1 + \cdots + v_n.

In words, the value of the whole is simply the sum of the values of the parts, regardless of their sizes. But if σ\sigma is to have the “absent elements” property, this won’t do. (Intuitively, if p i=0p_i = 0 then we shouldn’t count v iv_i in the sum, because the iith element isn’t actually there.) So we’d better modify this example slightly, instead taking

σ(p,v)=∑ i:p i>0v i. \sigma(\mathbf{p}, \mathbf{v}) = \sum_{i \,:\, p_i \gt 0} v_i.

This function (or rather, sequence of functions) does have the “absent elements” property.

Continuity in positive probabilities  Finally, we ask that for each v∈(0,∞) n\mathbf{v} \in (0, \infty)^n, the function σ(−,v)\sigma(-, \mathbf{v}) is continuous on the interior of the simplex Δ n\Delta_n, that is, continuous over those probability distributions p\mathbf{p} such that p 1,…,p n>0p_1, \ldots, p_n \gt 0.

Why only over the interior of the simplex? Basically because of natural examples of σ\sigma like the one just given, which is continuous on the interior of the simplex but not the boundary. Generally, it’s sometimes useful to make a sharp, discontinuous distinction between the cases p i>0p_i \gt 0 (presence) and p i=0p_i = 0 (absence).

 

Arrow’s famous theorem states that a few apparently mild conditions on a voting system are, in fact, mutually contradictory. The mild conditions above are not mutually contradictory. In fact, there’s a one-parameter family σ q\sigma_q of functions each of which satisfies these conditions. For real q≠1q \neq 1, the definition is

σ q(p,v)=(∑ i:p i>0p i qv i 1−q) 1/(1−q). \sigma_q(\mathbf{p}, \mathbf{v}) = \Bigl( \sum_{i \,:\, p_i \gt 0} p_i^q v_i^{1 - q} \Bigr)^{1/(1 - q)}.

For instance, σ 0\sigma_0 is the example of σ\sigma given above.

The formula for σ q\sigma_q is obviously invalid at q=1q = 1, but it converges to a limit as q→1q \to 1, and we define σ 1(p,v)\sigma_1(\mathbf{p}, \mathbf{v}) to be that limit. Explicitly, this gives

σ 1(p,v)=∏ i:p i>0(v i/p i) p i. \sigma_1(\mathbf{p}, \mathbf{v}) = \prod_{i \,:\, p_i \gt 0} (v_i/p_i)^{p_i}.

In the same way, we can define σ −∞\sigma_{-\infty} and σ ∞\sigma_\infty as the appropriate limits:

σ −∞(p,v)=max i:p i>0v i/p i,σ ∞(p,v)=min i:p i>0v i/p i. \sigma_{-\infty}(\mathbf{p}, \mathbf{v}) = \max_{i \,:\, p_i \gt 0} v_i/p_i, \qquad \sigma_{\infty}(\mathbf{p}, \mathbf{v}) = \min_{i \,:\, p_i \gt 0} v_i/p_i.

And it’s easy to check that for each q∈[−∞,∞]q \in [-\infty, \infty], the function σ q\sigma_q satisfies all the natural conditions listed above.

These functions σ q\sigma_q might be unfamiliar to you, but they have some special cases that are quite well-explored. In particular:

• Suppose you’re in a situation where the elements don’t have “sizes”. Then it would be natural to take p\mathbf{p} to be the uniform distribution u n=(1/n,…,1/n)\mathbf{u}_n = (1/n, \ldots, 1/n). In that case, σ q(u n,v)=const⋅(∑v i 1−q) 1/(1−q), \sigma_q(\mathbf{u}_n, \mathbf{v}) = const \cdot \bigl( \sum v_i^{1 - q} \bigr)^{1/(1 - q)}, where the constant is a certain power of nn. When q≤0q \leq 0, this is exactly a constant times ‖v‖ 1−q\|\mathbf{v}\|_{1 - q}, the (1−q)(1 - q)-norm of the vector v\mathbf{v}.

• Suppose you’re in a situation where the elements don’t have “values”. Then it would be natural to take v\mathbf{v} to be 1=(1,…,1)\mathbf{1} = (1, \ldots, 1). In that case, σ q(p,1)=(∑p i q) 1/(1−q). \sigma_q(\mathbf{p}, \mathbf{1}) = \bigl( \sum p_i^q \bigr)^{1/(1 - q)}. This is the quantity that ecologists know as the Hill number of order qq and use as a measure of biological diversity. Information theorists know it as the exponential of the Rényi entropy of order qq, the special case q=1q = 1 being Shannon entropy. And actually, the general formula for σ q\sigma_q is very closely related to Rényi relative entropy (which Wikipedia calls Rényi divergence).

Anyway, the big — and as far as I know, new — result is:

Theorem  The functions σ q\sigma_q are the only functions σ\sigma with the seven properties above.

So although the properties above don’t seem that demanding, they actually force our notion of “aggregate value” to be given by one of the functions in the family (σ q) q∈[−∞,∞](\sigma_q)_{q \in [-\infty, \infty]}. And although I didn’t even mention the notions of diversity or entropy in my justification of the axioms, they come out anyway as special cases.

I covered all this yesterday in the tenth and penultimate installment of the functional equations course that I’m giving. It’s written up on pages 38–42 of the notes so far. There you can also read how this relates to more realistic measures of biodiversity than the Hill numbers. Plus, you can see an outline of the (quite substantial) proof of the theorem above.

The ability to learn tasks in a sequential fashion is crucial to the development of artificial intelligence. Until now neural networks have not been capable of this and it has been widely thought that catastrophic forgetting is an inevitable feature of connectionist models. We show that it is possible to...

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Florida edition.

As punishment, four corrections officers — John Fan Fan, Cornelius Thompson, Ronald Clarke and Edwina Williams — kept Rainey in that shower for two full hours. Rainey was heard screaming "Please take me out! I can’t take it anymore!” and kicking the shower door. Inmates said prison guards laughed at Rainey and shouted "Is it hot enough?"

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