Subject: Re: [AUDITORY] Logan's theorem - a challenge From: Douglas Scott <jdmusictuition@xxxxxxxx> Date: Sun, 26 Sep 2021 12:08:43 -0400--00000000000083fa8005cce832ab Content-Type: text/plain; charset="UTF-8" Content-Transfer-Encoding: quoted-printable Hi Alain This is quite an interesting concept. I have been working on something similar using Zipfian distributions which involves rank ordering signals, which obviously destroys a lot of information. But if you "play out" this process across multiple scales, it is perfectly possible to put a sine wave in and get a bell curve out. So, in principle, you can use this technique to reconstruct any signal. The connection here is that the pink noise profile associated with Zipf's law is strongly associated with band-limited signals. In fact, the ubiquity of that kind of distribution likely has something to do with necessary band-limiting that occurs with any sampled signal. Here one can consider what happens inside the brain as a sort of corollary to the Nyquist theorem, because even though you only need a finite number of samples to describe a signal with a given maximum frequency component, the number of perspectives of that signal is effectively infinite (short of a universal theory of everything in physics). So the perceptual mechanism plus signal system taken as a whole is necessarily band-limited in that sense. Zipf's law is a kind of projective optimum for that kind of system, and it makes sense that biological systems have a strong evolutionary impulse to arrange itself in the most efficient way. Obviously it's more complicated than that, but that's a simple way to understand it. The kicker of thinking about minds in this way is that we can dispense with problematic concepts such as real existences of objects and things as the basic building blocks of cognition. A simple neural network such as that of C. Elegans doesn't need to build a representation of the world, that representation builds itself simply in the gradients across the signals that it is receiving, and how those gradients change as the worm moves in its surroundings. In evolutionary terms it then becomes a comparatively simple matter of selecting for genotypes that optimize the sensitivity to things in the environment that allows the worm to succeed. It gives a nice model of cognition that allows human thought to fit into a continuum, because such a system inherently scales: More neurons =3D more opportunitie= s discrimination. It helps to explain a lot of other things such as the plasticity of the brain in contrast to the specialisation of brain regions for specific tasks. It also goes a long way towards making the difference between current generation AI and human cognition obvious. AI training starts with "pre-coded" targets as sort of "victory conditions". That's not a good analogue for real brains, where there is no real victory condition. One could say that minds are not teleologically constructed in that way: They are not constructed to see one thing or another, but optimally discriminate between things. So it's not the case that language, for example, obeys Zipf's law, it is that our understanding of things tends towards Zipf's law because that is the optimum way for us to arrange objects in our understanding (or items in our perception). One thing that I have found in trying to code this sort of thing is that, while it seems quite straightforward in an analogue "mind", it scales poorly in a digital sense. That's simply because a wet and squishy mind has far more states that it can respond to than simply a 1/0 electrical impulse. There are chemical, mechanical, thermodynamic and even quantum effects that affect even a single neuron, and all of that can be roped into making discriminations about internal and external states that you need to navigate an environment. In that sense, to bring it back to Logan's theorem, one way of thinking about it may be that each element of a cognitive system could be considered perfectly band-limited within a narrowly defined domain, and would be useful to follow elements deeper into the system to that extent. Learning can then be understood as refining the domains to which these elements are applied to. Anyway, long rant, but hopefully you can find something useful there. Doug On Sun, 26 Sept 2021 at 05:47, Alain de Cheveigne < alain.de.cheveigne@xxxxxxxx> wrote: > Hi all, > > Here=E2=80=99s a challenge for the young nimble minds on this list, and t= he old > and wise. > > Logan=E2=80=99s theorem states that a signal can be reconstructed from it= s zero > crossings, to a scale, as long as the spectral representation of that > signal is less than an octave wide. It sounds like magic given that zero > crossing information is so crude. How can the full signal be recovered fr= om > a sparse series of time values (with signs but no amplitudes)? > =E2=80=9CBand-limited=E2=80=9D is clearly a powerful assumption. > > Why is this of interest in the auditory context? The band-limited premis= e > is approximately valid for each channel of the cochlear filterbank > (sometimes characterized as a 1/3 octave filter). While cochlear > transduction is non-linear, Logan=E2=80=99s theorem suggests that any inf= ormation > lost due to that non-linearity can be restored, within each channel. If s= o, > cochlear transduction is =E2=80=9Ctransparent=E2=80=9D, which is encourag= ing for those who > like to speculate about neural models of auditory processing. An algorith= m > applicable to the sound waveform can be implemented by the brain with > similar results, in principle. > > Logan=E2=80=99s theorem has been invoked by David Marr for vision and sev= eral > authors for hearing (some refs below). The theorem is unclear as to how t= he > original signal should be reconstructed, which is an obstacle to > formulating concrete models, but in these days of machine learning it mig= ht > be OK to assume that the system can somehow learn to use the information, > granted that it=E2=80=99s there. The hypothesis has far-reaching implica= tions, for > example it implies that spectral resolution of central auditory processin= g > is not limited by peripheral frequency analysis (as already assumed by fo= r > example phase opponency or lateral inhibitory hypotheses). > > Before venturing further along this limb, it=E2=80=99s worth considering = some > issues. First, Logan made clear that his theorem only applies to a > perfectly band-limited signal, and might not be =E2=80=9Capproximately va= lid=E2=80=9D for a > signal that is =E2=80=9Capproximately band-limited=E2=80=9D. No practica= l signal is > band-limited, if only because it must be time limited, and thus the theor= em > might conceivably not be applicable at all. On the other hand, half-wave > rectification offers much richer information than zero crossings, so > perhaps the end result is valid (information preserved) even if the theor= em > is not applicable stricto sensu. Second, there are many other > imperfections such as adaptation, stochastic sampling to a spike-based > representation, and so on, that might affect the usefulness of the > hypothesis. > > The challenge is to address some of these loose ends. For example: > (1) Can the theorem be extended to make use of a halfwave-rectified signa= l > rather than zero crossings? Might that allow it to be applicable to > practical time-limited signals? > (2) What is the impact of real cochlear filter characteristics, > adaptation, or stochastic sampling? > (3) In what sense can one say that the acoustic signal is "available=E2= =80=9D to > neural signal processing? What are the limits of that concept? > (4) Can all this be formulated in a way intelligible by non-mathematical > auditory scientists? > > This is the challenge. The reward is - possibly - a better understanding > of how our brain hears the world. > > Alain > > --- > Logan BF, JR. (1977) Information in the zero crossings of bandpass > signals. Bell Syst. Tech. J. 56:487=E2=80=93510. > > Marr, D. (1982) VISION - A Computational Investigation into the Human > Representation and Processing of Visual Information. W.H. Freeman and Co, > republished by MIT press 2010. > > Heinz, M.G., Swaminathan J. (2009) Quantifying Envelope and Fine-Structur= e > Coding in Auditory Nerve Responses to Chimaeric Speech, JARO 10: 407=E2= =80=93423 > DOI: 10.1007/s10162-009-0169-8. > > Shamma, S, Lorenzi, C (2013) On the balance of envelope and temporal fine > structure in the encoding of speech in the early auditory system, J. > Acoust. Soc. Am. 133, 2818=E2=80=932833. > > Parida S, Bharadwaj H, Heinz MG (2021) Spectrally specific temporal > analyses of spike-train responses to complex sounds: A unifying framework= . > PLoS Comput Biol 17(2): e1008155. > https://doi.org/10.1371/journal.pcbi.1008155 > > de Cheveign=C3=A9, A. (in press) Harmonic Cancellation, a Fundamental of > Auditory Scene Analysis. Trends in Hearing (https://psyarxiv.com/b8e5w/). --00000000000083fa8005cce832ab Content-Type: text/html; charset="UTF-8" Content-Transfer-Encoding: quoted-printable <div dir=3D"ltr"><div>Hi Alain</div><div><br></div>This is quite an interes= ting concept. I have been working on something similar using Zipfian=C2=A0d= istributions which involves rank ordering signals, which obviously destroys= a lot of information. But if you "play out" this process across = multiple scales, it is perfectly possible to put a sine wave in and get a b= ell curve out. So, in principle, you can use this technique to reconstruct = any signal. The connection here is that the pink noise profile associated w= ith Zipf's law is strongly associated with band-limited signals. In fac= t, the ubiquity of that kind of distribution likely has something to do wit= h necessary band-limiting that occurs with any sampled signal. Here one can= consider what happens inside the brain as a sort of corollary=C2=A0to the = Nyquist theorem, because even though you only need a finite number of sampl= es to describe a signal with a given maximum frequency component, the numbe= r of perspectives of that signal is effectively infinite (short of a univer= sal theory of everything in physics). So the perceptual mechanism plus sign= al system taken as a whole is necessarily band-limited in that sense. Zipf&= #39;s law is a kind of projective optimum for that kind of system, and it m= akes=C2=A0sense that=C2=A0biological=C2=A0systems have a strong evolutionar= y=C2=A0impulse to arrange itself in the most efficient=C2=A0way. Obviously = it's more complicated than that, but that's a simple way to underst= and it.<div><br></div><div>The kicker of thinking about minds in this way i= s that we can dispense with problematic concepts such as real existences of= objects and things as the basic building blocks of cognition. A simple neu= ral network such as that of C. Elegans doesn't need to build a represen= tation of the world, that representation builds itself simply in the gradie= nts across the signals that it is receiving, and how those gradients change= as the worm moves in its surroundings. In evolutionary terms it then becom= es a comparatively simple matter of selecting for genotypes that optimize t= he sensitivity to things in the environment that allows the worm to succeed= . It gives a nice model of cognition that allows human thought to fit into = a continuum, because such a system inherently scales: More neurons =3D more= opportunities discrimination. It helps to explain a lot of other things su= ch as the plasticity of the brain in contrast to the specialisation of brai= n regions for specific tasks.=C2=A0</div><div><br></div><div>It also=C2=A0g= oes=C2=A0a=C2=A0long way towards making the difference between current gene= ration AI and human cognition obvious. AI training starts with "pre-co= ded" targets as sort of "victory conditions". That's not= a good analogue for real brains, where there is no real victory condition.= One could say that minds are not teleologically constructed in that way: T= hey are not constructed to see one thing or another, but optimally discrimi= nate=C2=A0between things. So it's not the case that language, for examp= le, obeys Zipf's law, it is that our understanding of things tends towa= rds Zipf's law because that is the optimum way for us to arrange object= s in our understanding (or items in our perception).</div><div><br></div><d= iv>One thing that I have found in trying to code this sort of thing is that= , while it seems quite straightforward in an analogue "mind", it = scales poorly in a digital sense. That's simply because a wet and squis= hy mind has far more states that it can respond to than simply a 1/0 electr= ical=C2=A0impulse. There are chemical, mechanical, thermodynamic and even q= uantum effects that affect even a single neuron, and all of that can be rop= ed into making discriminations=C2=A0about=C2=A0internal and external states= that you need to navigate an environment.</div><div><br></div><div>In that= sense, to bring it back to Logan's theorem, one way of thinking about = it may be that each element of a cognitive system could be considered perfe= ctly band-limited within a narrowly defined domain, and would be useful to = follow elements deeper into the system to that extent. Learning can then be= understood as refining the domains to which these elements are applied to.= </div><div><br></div><div>Anyway, long rant, but hopefully you can find som= ething useful there.</div><div><br></div><div>Doug</div><div><div><br></div= ><div><br></div></div></div><br><div class=3D"gmail_quote"><div dir=3D"ltr"= class=3D"gmail_attr">On Sun, 26 Sept 2021 at 05:47, Alain de Cheveigne <= ;<a href=3D"mailto:alain.de.cheveigne@xxxxxxxx">alain.de.cheveigne@xxxxxxxx= l.eu</a>> wrote:<br></div><blockquote class=3D"gmail_quote" style=3D"mar= gin:0px 0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1= ex">Hi all,<br> <br> Here=E2=80=99s a challenge for the young nimble minds on this list, and the= old and wise.<br> <br> Logan=E2=80=99s theorem states that a signal can be reconstructed from its = zero crossings, to a scale, as long as the spectral representation of that = signal is less than an octave wide.=C2=A0 It sounds like magic given that z= ero crossing information is so crude. How can the full signal be recovered = from a sparse series of time values (with signs but no amplitudes)?=C2=A0 = =E2=80=9CBand-limited=E2=80=9D is clearly a powerful assumption.<br> <br> Why is this of interest in the auditory context?=C2=A0 The band-limited pre= mise is approximately valid for each channel of the cochlear filterbank (so= metimes characterized as a 1/3 octave filter).=C2=A0 While cochlear transdu= ction is non-linear, Logan=E2=80=99s theorem suggests that any information = lost due to that non-linearity can be restored, within each channel. If so,= cochlear transduction is =E2=80=9Ctransparent=E2=80=9D, which is encouragi= ng for those who like to speculate about neural models of auditory processi= ng. An algorithm applicable to the sound waveform can be implemented by the= brain with similar results, in principle.=C2=A0 <br> <br> Logan=E2=80=99s theorem has been invoked by David Marr for vision and sever= al authors for hearing (some refs below). The theorem is unclear as to how = the original signal should be reconstructed, which is an obstacle to formul= ating concrete models, but in these days of machine learning it might be OK= to assume that the system can somehow learn to use the information, grante= d that it=E2=80=99s there.=C2=A0 The hypothesis has far-reaching implicatio= ns, for example it implies that spectral resolution of central auditory pro= cessing is not limited by peripheral frequency analysis (as already assumed= by for example phase opponency or lateral inhibitory hypotheses).<br> <br> Before venturing further along this limb, it=E2=80=99s worth considering so= me issues.=C2=A0 First, Logan made clear that his theorem only applies to a= perfectly band-limited signal, and might not be =E2=80=9Capproximately val= id=E2=80=9D for a signal that is =E2=80=9Capproximately band-limited=E2=80= =9D.=C2=A0 No practical signal is band-limited, if only because it must be = time limited, and thus the theorem might conceivably not be applicable at a= ll.=C2=A0 On the other hand, half-wave rectification offers much richer inf= ormation than zero crossings, so perhaps the end result is valid (informati= on preserved) even if the theorem is not applicable stricto sensu.=C2=A0 Se= cond, there are many other imperfections such as adaptation, stochastic sam= pling to a spike-based representation, and so on, that might affect the use= fulness of the hypothesis.<br> <br> The challenge is to address some of these loose ends. For example:<br> (1) Can the theorem be extended to make use of a halfwave-rectified signal = rather than zero crossings? Might that allow it to be applicable to practic= al time-limited signals?<br> (2) What is the impact of real cochlear filter characteristics, adaptation,= or stochastic sampling?=C2=A0 <br> (3) In what sense can one say that the acoustic signal is "available= =E2=80=9D to neural signal processing?=C2=A0 What are the limits of that co= ncept?<br> (4) Can all this be formulated in a way intelligible by non-mathematical au= ditory scientists?<br> <br> This is the challenge.=C2=A0 The reward is - possibly - a better understand= ing of how our brain hears the world.<br> <br> Alain<br> <br> ---<br> Logan BF, JR. (1977) Information in the zero crossings of bandpass signals.= Bell Syst. Tech. J. 56:487=E2=80=93510.<br> <br> Marr, D. (1982) VISION - A Computational Investigation into the Human Repre= sentation and Processing of Visual Information. W.H. Freeman and Co, republ= ished by MIT press 2010.<br> <br> Heinz, M.G., Swaminathan J. (2009) Quantifying Envelope and Fine-Structure = Coding in Auditory Nerve Responses to Chimaeric Speech, JARO 10: 407=E2=80= =93423<br> DOI: 10.1007/s10162-009-0169-8.<br> <br> Shamma, S, Lorenzi, C (2013) On the balance of envelope and temporal fine s= tructure in the encoding of speech in the early auditory system, J. Acoust.= Soc. Am. 133, 2818=E2=80=932833.<br> <br> Parida S, Bharadwaj H, Heinz MG (2021) Spectrally specific temporal analyse= s of spike-train responses to complex sounds: A unifying framework. PLoS Co= mput Biol 17(2): e1008155. <a href=3D"https://doi.org/10.1371/journal.pcbi.= 1008155" rel=3D"noreferrer" target=3D"_blank">https://doi.org/10.1371/journ= al.pcbi.1008155</a><br> <br> de Cheveign=C3=A9, A. (in press) Harmonic Cancellation, a Fundamental of Au= ditory Scene Analysis. Trends in Hearing (<a href=3D"https://psyarxiv.com/b= 8e5w/" rel=3D"noreferrer" target=3D"_blank">https://psyarxiv.com/b8e5w/</a>= ).</blockquote></div> --00000000000083fa8005cce832ab--