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<meta name="author" content="Ross W. Gayler">
<meta name="dcterms.date" content="2023-06-15">
<title>Thinking about Vector Symbolic Architectures</title>
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<div class="slides">
<section id="title-slide" class="quarto-title-block center">
<h1 class="title">Thinking about Vector Symbolic Architectures</h1>
<div class="quarto-title-authors">
<div class="quarto-title-author">
<div class="quarto-title-author-name">
<a href="www.rossgayler.com">Ross W. Gayler</a> <a href="https://orcid.org/0000-0003-4679-585X" class="quarto-title-author-orcid"> <img src="data:image/png;base64,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"></a>
</div>
<div class="quarto-title-author-email">
<a href="mailto:[email protected]">[email protected]</a>
</div>
<p class="quarto-title-affiliation">
Independent Researcher
</p>
</div>
</div>
<p class="date">2023-06-15</p>
</section>
<section>
<section id="introduction" class="title-slide slide level1 center">
<h1>Introduction</h1>
</section>
<section id="motivation" class="slide level2">
<h2>Motivation</h2>
<ul>
<li>How far would you get with knowing the definitions of the VSA operators and <em>nothing</em> else?
<ul>
<li>Like having axioms and no other maths knowledge</li>
</ul></li>
<li>Everyone has a web of auxiliary <em>beliefs</em> around VSA:
<ul>
<li>About VSA and relationships to other things they know</li>
<li>Used to reason about their VSA knowledge and its implications, so is central to applying and extending VSA</li>
</ul></li>
<li>The objective is that the auxiliary beliefs should be <em>productive</em> for applying and extending VSA</li>
</ul>
</section>
<section id="limitations" class="slide level2">
<h2>Limitations</h2>
<p>Auxiliary beliefs are:</p>
<ul>
<li>Not necessarily true</li>
<li>Not necessarily even true or false (e.g. metaphors)</li>
<li>Not necessarily coherent</li>
<li>Not stable over time. Think of them as evolving frameworks.</li>
<li>Likely to be idiosyncratic (If they were canonical they would be VSA theory)</li>
</ul>
<p>Think of them as gambles. You are betting that they will be more productive than alternatives you might entertain.</p>
</section>
<section id="disorganisation-of-the-talk" class="slide level2">
<h2>(dis)Organisation of the talk</h2>
<ul>
<li>I am talking about my conceptual framework
<ul>
<li><p>I have no idea what yours is, because we don’t discuss it</p></li>
<li><p>This may be painfully obvious to you. I apologise.</p></li>
</ul></li>
<li>Framework is densely connected web of interrelated points
<ul>
<li><p>No simple, logical, explanatory path through that web</p></li>
<li><p>Any path is necessarily a random-ish ramble. I apologise.</p></li>
</ul></li>
<li>Pick some apparently salient points, wander in their neighbourhoods, much hand waving</li>
</ul>
</section></section>
<section>
<section id="analogue-computer-wire-interpretation" class="title-slide slide level1 center">
<h1>Analogue computer wire interpretation</h1>
</section>
<section id="composite-label-and-magnitude" class="slide level2">
<h2>Composite label and magnitude</h2>
<p><strong>Q</strong>: What is a (hyper)vector? <strong>A</strong>: Direction + scalar magnitude</p>
<ul>
<li>Electric analogue computers represent values on wires. The voltage is the magnitude of the signal, the “meaning” of the signal is a label on the wire.</li>
<li>Interpret the vector magnitude as the signal magnitude.</li>
<li>Interpret the vector direction as the signal label.</li>
<li>Labels are composite - (de)composed by VSA operators</li>
<li>Labels (wires) can be created on the fly</li>
</ul>
<p>Think of VSA systems as analogue computers</p>
</section>
<section id="labels-can-encode-values" class="slide level2">
<h2>Labels can encode values</h2>
<p>Is it too limiting for values to be scalar magnitudes? What if I want structured values?</p>
<ul>
<li>Composite labels can encode values, e.g.<br>
<span class="math inline">\(colour \times red\)</span> or <span class="math inline">\(height \times encode(180)\)</span>
<ul>
<li>Interpret the label as a predicate</li>
<li>Interpret magnitude as truth value or degree of support</li>
</ul></li>
<li>To avoid -ve magnitude, put the sign in the label: <span class="math inline">\(-(a)\)</span>
<ul>
<li>Use unary additive and multiplicative inverse operators</li>
</ul></li>
<li>May need mechanism (e.g. RELU in cleanup) to enforce this</li>
</ul>
</section>
<section id="labels-can-encode-complicated-values" class="slide level2">
<h2>Labels can encode complicated values</h2>
<p><strong>Slogan</strong>: <u>Everything is just a vector</u></p>
<ul>
<li>In GOFAI data structures, slots/fields/keys are atomic symbols: <span class="math inline">\(colour\)</span>, <span class="math inline">\(height\)</span>, …</li>
<li>In VSA labels can be arbitrarily complicated, e.g.<br>
<span class="math inline">\(go\_to\_kitchen\_and\_look\_in\_fridge \times beer\)</span><br>
<span style="font-size: 60%">(where <span class="math inline">\(go\_to\_kitchen\_and\_look\_in\_fridge\)</span> represents an executable sensorimotor program)</span><br>
is equivalent to the agent’s degree of belief that there is beer in the refrigerator.</li>
</ul>
</section>
<section id="label-symmetries-and-equivalences" class="slide level2">
<h2>Label symmetries and equivalences</h2>
<p><strong>Slogan</strong>: <u>Every vector is just a value</u></p>
<ul>
<li>VSA “sees” the vector value, not the sequence of operations
<ul>
<li>Different sequences of operations can be equivalent, e.g.
<ul>
<li><span class="math inline">\(a + b = b + a\)</span> (if bundling is commutative)</li>
<li>Circular convolution binding of <span class="math inline">\(d\)</span>-dimensional vectors is equal to the bundle of <span class="math inline">\(d\)</span>-many Hadamard bindings of permutations of the arguments</li>
</ul></li>
</ul></li>
<li>“noise” can make unequal values effectively equal, e.g.<br>
<span class="math inline">\(a + b + \cdots + y \approx a + b + \cdots + z\)</span> (bundling capacity)</li>
</ul>
</section>
<section id="occams-hypervectors" class="slide level2">
<h2>Occam’s hypervectors</h2>
<ul>
<li>Start with the atomic vectors in their vector space</li>
<li>Repeatedly apply the operators to generate expressions of increasing length</li>
<li>Crowding of result vectors increases with expression length</li>
<li>Every sufficiently long expression will be approximately equal to some shorter expression</li>
<li>Implies a form of Occam’s Razor: The system treats any value as effectively arising from the simplest expression that approximately generates that value.</li>
</ul>
</section></section>
<section>
<section id="similarities" class="title-slide slide level1 center">
<h1>Similarities</h1>
</section>
<section id="angular-similarity-and-distance" class="slide level2">
<h2>Angular similarity and distance</h2>
<ul>
<li>Similarity is central to reasoning about VSA</li>
<li>Angle between vectors (equivalent to dot product) is appropriate because it respects the arguments as vectors
<ul>
<li><p>Vectors are defined with respect to an origin</p></li>
<li><p>Distances between points are invariant to translation</p></li>
<li><p>Angle between vectors to the same points are not invariant to translation of the origin</p></li>
</ul></li>
<li>Distance and angle can be equivalent if points constrained to a fixed origin, e.g. Hamming distance for binary vectors</li>
</ul>
</section>
<section id="a-view-from-the-north-pole" class="slide level2">
<h2>A view from the north pole</h2>
<ul>
<li>Imagine the hypervector representing the state of a VSA system is the north pole and you are standing there</li>
<li>Look at the locations of random hypervectors. They are almost all very near the equator (quasiorthogonality)</li>
<li>Similarity is defined on pairs of vectors. Consider pairs related by transforms available to the system
<ul>
<li><p>Pairs within the equatorial belt (the vast majority) have no impact on similarity relative to pole (current state)</p></li>
<li><p>Angular similarity can’t be only driver of system dynamics</p></li>
</ul></li>
</ul>
</section>
<section id="edit-distance-for-bindingpermutation" class="slide level2">
<h2>Edit distance for binding/permutation</h2>
<ul>
<li>Angular similarity is essentially about transforms within a hemi(hyper)sphere</li>
<li>Angular similarity is driven by bundling</li>
<li>Binding and permutation are equatorial belt transforms</li>
<li>Something like an edit distance might be useful for selecting between equatorial transforms
<ul>
<li><p>Every destination is only one transform away if we allow arbitrary binding and permutation</p></li>
<li><p>Restrict to available permutations and <a href="https://en.wikipedia.org/wiki/Currying">curried</a> bindings</p></li>
</ul></li>
</ul>
</section></section>
<section>
<section id="estimating-latent-reality" class="title-slide slide level1 center">
<h1>Estimating latent reality</h1>
</section>
<section id="latent-reality-and-hypervectors" class="slide level2">
<h2>Latent reality and hypervectors</h2>
<ul>
<li>We concentrate on the observable hypervectors</li>
<li>Change our focus to unobservable reality
<ul>
<li>Inspired by being an applied statistician</li>
<li>Much of statistics is about inferring the state of an unobservable reality from observable measurements</li>
</ul></li>
<li>Hypervectors as observable realisations of measurements of unobservable (latent) reality
<ul>
<li>Try to explain the role of randomness in VSA</li>
</ul></li>
</ul>
</section>
<section id="elements-as-random-measurements" class="slide level2">
<h2>Elements as random measurements</h2>
<ul>
<li>No direct access to reality (mediated by measurements)</li>
<li>Imagine looking at an object
<ul>
<li>Random whether an atom reflects a photon to your eye</li>
<li>Random whether a photoreceptor is in the path</li>
</ul></li>
<li>Ultradimensional vector of potential measurements
<ul>
<li>Hypervector is a random sample of them: <span class="math inline">\(\phi_i(x)\)</span>
<ul>
<li><span class="math inline">\(\phi_i\)</span> is a function from reality <span class="math inline">\(x\)</span> to VSA base field</li>
</ul></li>
<li>Want to infer the reality despite randomness of sample</li>
</ul></li>
</ul>
</section>
<section id="properties-of-random-measurements" class="slide level2">
<h2>Properties of random measurements</h2>
<ul>
<li>VSA doesn’t “know” <span class="math inline">\(\phi_i\)</span> (knows the value, not the function)</li>
<li>Want measurements to be individually and collectively informative about <span class="math inline">\(x\)</span> (reality)
<ul>
<li><p>Need to depend on properties of <span class="math inline">\(x\)</span> we want to capture</p></li>
<li><p>Need to be independent conditional on <span class="math inline">\(x\)</span> (hash of <span class="math inline">\(x\)</span>)</p></li>
</ul></li>
<li>Information is carried by covariation of <span class="math inline">\(\phi_i(x)\)</span> induced by variation in <span class="math inline">\(x\)</span></li>
<li>No measurement is privileged (implies robustness, distributed representation, and unordered representation)</li>
</ul>
</section>
<section id="measurements-as-constraints" class="slide level2">
<h2>Measurements as constraints</h2>
<p>Imagine the VSA base field is binary:</p>
<ul>
<li>Each specific <span class="math inline">\(\phi_i(x)\)</span> is consistent with a set of half the possible latent realities</li>
<li>A set of hypervector elements specifies the intersection of those sets
<ul>
<li>Each element narrows the set of consistent realities</li>
</ul></li>
</ul>
<p>Interpret hypervector as specifying a set of indistinguishable realities rather than being a representation of a single reality</p>
</section>
<section id="possible-implicationsextensions" class="slide level2">
<h2>Possible implications/extensions</h2>
<ul>
<li>More feasible search over reality because of the inexactness due to indistinguishability?</li>
<li>Only need sufficient dimensionality to make the distinctions we need
<ul>
<li>Optimisation of dimensionality</li>
<li>Dynamic dimensionality?</li>
</ul></li>
<li>Possibility of dynamic constraints
<ul>
<li>Effectively adding or removing measurements</li>
<li>Dynamically creating new measurements on the fly</li>
</ul></li>
</ul>
</section></section>
<section>
<section id="statistical-interpretation-of-algebraic-terms" class="title-slide slide level1 center">
<h1>Statistical interpretation of algebraic terms</h1>
</section>
<section id="statistical-data-structure" class="slide level2">
<h2>Statistical data structure</h2>
<ul>
<li>Standard statistical data structure is a matrix
<ul>
<li>Rows are cases/observations</li>
<li>Columns are variables/features (predictors and outcomes)</li>
</ul></li>
<li>Columns structurally orthogonal in data space (bases)</li>
<li>Features are often single columns (scalar values)</li>
<li>Features can be groups of columns (e.g. one-hot encoding)</li>
<li>Statistical modelling is about exploiting covariation of feature values induced by variation over cases (familiar?)</li>
</ul>
</section>
<section id="terms-as-rotated-features" class="slide level2">
<h2>Terms as rotated features</h2>
<ul>
<li>In statistical modelling, we could rotate the data space
<ul>
<li>Features are spread over columns</li>
<li>Works if rotated features are orthogonal in the data space</li>
</ul></li>
<li>VSA uses regression to extract predictions from hypervectors
<ul>
<li>Hypervectors often expressible as sum of algebraic terms</li>
<li>Algebraic terms are often orthogonal (rotated features)
<ul>
<li>Simple terms represent main effects</li>
<li>Bindings represent interactions</li>
</ul></li>
</ul></li>
</ul>
</section>
<section id="possible-implications" class="slide level2">
<h2>Possible implications</h2>
<ul>
<li>Operation of VSA regression/classification systems can be understood/analysed with respect to terms in hypervector
<ul>
<li>E.g. Integer Echo State Network builds standard sequence representation (Interpretable as set of lagged inputs)</li>
</ul></li>
<li>Representations can be designed to achieve objectives
<ul>
<li>What features needed for standard regression?</li>
<li>Create algebraic terms that implement those features
<ul>
<li>E.g. Epileptic Seizure Challenge needed interactions of time-series features with time of day (bindings)</li>
</ul></li>
</ul></li>
</ul>
</section></section>
<section>
<section id="indices-and-permutation" class="title-slide slide level1 center">
<h1>Indices and permutation</h1>
</section>
<section id="element-indices-as-unique-labels" class="slide level2">
<h2>Element indices as unique labels</h2>
<ul>
<li>Computer people tend to think of vector indices as consecutive integers: <span class="math inline">\(a_i\)</span> where <span class="math inline">\(i = 1, 2, \ldots\)</span>
<ul>
<li>This imposes more structure than necessary</li>
</ul></li>
<li>Indices only need to be unique : <span class="math inline">\(i = sad, bee, hot, \ldots\)</span></li>
<li>Indices do <em>not</em> need to be ordered
<ul>
<li>Ordering convenient for 2D electronic implementation</li>
<li>Ordering is an imposition for 3D neural implementation</li>
</ul></li>
<li>Hypervector is a set of key-value pairs where the values are from the VSA base field (sound familiar?)</li>
</ul>
</section>
<section id="permutation-and-operators" class="slide level2">
<h2>Permutation and operators</h2>
<ul>
<li>It doesn’t make sense to talk of permuting an isolated hypervector (interpreted as set of key-value pairs) because it’s unordered</li>
<li>Makes sense to talk of permutation:
<ul>
<li>relative to another vector,</li>
<li>when they are being combined by an operator,</li>
<li>because it’s about tracking which elements are combined <span class="math display">\[\{a_{a1}, a_{a2}, \ldots\} + \rho\{b_{b1}, b_{b2}, \ldots\} = \{x_{a1.b2}, x_{a2.b3}, \ldots\}\]</span></li>
</ul></li>
</ul>
</section>
<section id="possible-implications-1" class="slide level2">
<h2>Possible implications</h2>
<ul>
<li>What makes a tensor product a tensor product is the pattern of combination of the elements of the arguments and the availability of that pattern to guide tensor operations (e.g. tensor contraction)</li>
<li>Key-value pairs (hypervector elements) can be represented and operated on as VSA hypervectors</li>
<li>Is it possible to self-embed?
<ul>
<li>Implement tensor product operations with VSA?</li>
<li>Have dynamic elements (add/remove elements)?</li>
</ul></li>
</ul>
</section></section>
<section id="archival-links" class="title-slide slide level1 center">
<h1>Archival links</h1>
<p>This presentation has been archived on <a href="https://zenodo.org/">Zenodo</a>:</p>
<p><a href="https://doi.org/10.5281/zenodo.8076677"><img data-src="https://zenodo.org/badge/DOI/10.5281/zenodo.8076677.svg" alt="DOI"></a> Video recording</p>
<p><a href="https://doi.org/10.5281/zenodo.8076707"><img data-src="https://zenodo.org/badge/DOI/10.5281/zenodo.8076707.svg" alt="DOI"></a> Slides (PDF)</p>
<p><a href="https://doi.org/10.5281/zenodo.8076736"><img data-src="https://zenodo.org/badge/DOI/10.5281/zenodo.8076736.svg" alt="DOI"></a> Source code of slides</p>
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<p>Midnight Sun Workshop on Vector Symbolic Architectures, June 15-16, 2023, Luleå, Sweden</p>
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