[VSAONLINE]. Starts in 30 minutes

[Evgeny Osipov]

… a brief reminder about VSAONLINE webinar which starts in 30 minutes. See description and link below.

Best

Evgeny

 

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Dear all,

 

Welcome to the next talk of Season 12 on VSAONLINE. Marco Angioli and Caio Vieirafrom Sapienza University of Rome, Italy will give a talk

”Efficient Hyperdimensional Computing with Modular Composite Representations / Hyle: An HLS Framework for Hyperdimensional Computing Accelerators on FPGAs”

 

Date: March 2,  2026

Time: 20:00 GMT 

Zoom: https://ltu-se.zoom.us/j/65564790287

 WEB: https://bit.ly/vsaonline

 

Abstract: The modular composite representation (MCR) is a computing model that represents information with high-dimensional integer vectors using modular arithmetic. Originally proposed as a generalization of the binary spatter code model, it aims to provide higher representational power while remaining a lighter alternative to models requiring high-precision components. Despite this potential, MCR has received limited attention. Systematic analyses of its trade-offs and comparisons with other models are lacking, sustaining the perception that its added complexity outweighs the improved expressivity. In this work, we revisit MCR by presenting its first extensive evaluation, demonstrating that it achieves a unique balance of capacity, accuracy, and hardware efficiency. Experiments measuring capacity demonstrate that MCR outperforms binary and integer vectors while approaching complex-valued representations at a fraction of their memory footprint. Evaluation on 123 datasets confirms consistent accuracy gains and shows that MCR can match the performance of binary spatter codes using up to 4x less memory. We investigate the hardware realization of MCR by showing that it maps naturally to digital logic and by designing the first dedicated accelerator. Evaluations on basic operations and 7 selected datasets demonstrate a speedup of up to 3 orders of magnitude and significant energy reductions compared to software implementation. When matched for accuracy against binary spatter codes, MCR achieves on average 3.08x faster execution and 2.68x lower energy consumption. These findings demonstrate that, although MCR requires more sophisticated operations than binary spatter codes, its modular arithmetic and higher per-component precision enable lower dimensionality. When realized with dedicated hardware, this results in a faster, more energy-efficient, and high-precision alternative to existing models.

 

[Tony Plate]

Thanks to Macro & Caio for the interesting talks on MCR and CGR today.

jcyn...@iu.edu asked the question: BSC and FHRR can both be regarded as vector spaces over fields. MCR and CGR are only vector spaces over fields if n is prime. The hardware advantage of using a power of 2 is intriguing. I'm wondering what you give up by working with a representation space that is a module over a ring, rather than a vector space over a field.

I think the answer to this question, at least to the first approximation, is that you give up basically nothing by working with a representation space that is a module over a ring, rather than a vector space over a field.

The reason is that the VSA operations over modular arithmetic don't utilize multiplication in the modulus space, so that lack of inverses there doesn't matter.

Google Gemini gives a decent summary of ring vs field for modular arithmetic:

Recall that the VSA binding operation using modulus arithmetic is actually addition, not multiplication.  There's a link to multiplication via the fact that addition of angles of complex numbers on the unit circle is equivalent to the multiplication of those complex numbers, but that's beside the point.  When we "bind" together 2 and 4 in mod-8 arithmetic, we add them to get 6, and we can "unbind" with 2 or 4 to get back the other.  The difference between a ring and a field only becomes important to multiplication in mod-8 space: the product of 2 and 4 mod 8 is 0.  Given the result 0 and one of the arguments, there's no way of reliably getting back the other argument (if we know x * 4 = 0 mod 8, then x could be 0 or 2 or 4 or 6.)

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[Cynamon, Josh]

Tony,

Thank you for replying to my question. I was asking it on the fly, so I wasn't sure how it might or might not be important. I appreciate you taking the time to think about it and send an email.

One thing I've thought some about is also the structure of the collection of representations (which is to say, the vectors themselves). I think that the collection of vectors and their bundling and binding operations in FHRR have more structure than the collection of vectors and their bundling and binding operations in BSC, but I could be wrong. Specifically, I thought each there was an identity for each operation and each vector had a unique inverse for each of addition and multiplication in FHRR, but not BSC. If that's correct, it's possible that there are useful theorems that would apply to one and not the other. Of course, even viewed as vector spaces, FHRR is a vector space over a field that's characteristic 0, so that might also confer some advantage (perhaps at the expense of some nice feature of being a vector space over a field of characteristic 2, as BSC is).

On the other hand, these could just be idle ramblings on my part.

Regards, in any case,

Josh (aka jcyn...@iu.edu)

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Cc: Leonid Mokrushin <leonid.m...@ericsson.com>; Cynamon, Josh <jcyn...@iu.edu>; Evgeny Osipov <evgeny...@ltu.se>; marco....@uniroma1.it <marco....@uniroma1.it>; crav...@inf.ufrgs.br <crav...@inf.ufrgs.br>
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[Jesper Olsen]

Interesting talk - I ran some software experiments with MCR (testing R=2**0,...,2**8).

I find that MCR performs well, but not necessarily better than BSCs
The R=1 case is nearly identical to BSC/bipolar, as expected.

I've tested this across text language ID and the inference examples from Tony’s 1990 paper.

Kanerva’s 'Mexican Dollar' analogy works perfectly for MCR when R=1, but accuracy degrades as R increases.
It seems that in the higher-resolution modular spaces, the unbinding operation doesn't strip away crosstalk
from the bundle as cleanly - the 'noise' left behind after unbinding is more disruptive.

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[Marco Angioli]

Hi Jesper,
Thanks for your interest in our work. I had a look at your code and results. Here are my observations:

• Regarding the inference example, your results show that MCR performs similarly to BSC. This is correct and expected, since the experiment involves very few bundling operations and may not be the best dataset to demonstrate the advantages of higher per-component precision.
  I am sharing a simple implementation of the classifier we used in our paper, based on the extended torchhd library. You can find the instructions for extending the library and running the model on different UCI datasets in the attached folder. 
  
  
• Regarding the Dollar of Mexico, your results clearly show that accuracy degrades for R > 1. This is true, but the reason is documented in this paper by Kenny Schlegel et al. (2022), "A Comparison of Vector Symbolic Architectures" (Section 2.5). The Dollar of Mexico in Kanerva's original formulation can only work if binding is self-inverse. This is not the case for MCR, where binding is modular addition and unbinding is modular subtraction. For models where binding is not self-inverse, instead of creating a single composite hypervector by binding the country records together, you should keep records stored separately and query them in two unbinding steps, with a cleanup in between. With this approach, the accuracy does not degrade and is in fact perfect for every R. It is worth noting that Kanerva himself proposed both formulations in his 1997 paper "Fully Distributed Representation": the single-vector approach (Solution 2u, which requires self-inverse binding) and the two-step approach (Solution 2c, which works with any VSA).  

I hope this is useful, and I am happy to discuss further. Thanks also to everyone for the interesting questions and discussion that originated from the talk.  

Best,

Marco

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[Jesper Olsen]

Hi Marco,

Many thanks for your explanation and the reference to Kanerva's 1997 paper.

I was indeed using the single-vector (solution 2u) approach which assumes self-inverse binding - which BSC
doesn't have for R>1. It works with solution 2c.

I’m also looking through the notebook you shared.

Cheers
Jesper

[Artem Hnilov]

Hello,

I used BSC hypervectors together with a Fuzzy-Pattern Tsetlin Machine for character-level text generation in the style of Shakespeare. What potential benefits might I get if I switch to MCR hypervectors?

Am I correct in understanding that the symbol position in the context can be encoded using different rotation speeds of the hypervector's MCR components?

Also, am I correct in understanding that, when binarizing the MCR representation before feeding it to the FPTM input, it is better to use a circular thermometer code (01111000)?

Thank you.

Here is the project with a text generation example:
https://github.com/BooBSD/Tsetlin.jl

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[Ross Gayler]

I would like to briefly expand on two points touched on in Tony's email:

1. TAP> Recall that the VSA binding operation using modulus arithmetic is actually addition, not multiplication.  There's a link to multiplication via the fact that addition of angles of complex numbers on the unit circle is equivalent to the multiplication of those complex numbers, but that's beside the point.
   
   One popular variant of complex-valued VSAs is to use phasors rather than unconstrained complex numbers (that is, complex numbers constrained to lie on the unit circle in the complex plane). The phase angles of the phasors are isomorphic to the reals modulo 2*pi and binding/unbinding are modular addition/subtraction of phase angles. The point I want to emphasise here is that the set of complex values constituting the set of possible phasors does *not* include the multiplicative zero value for complex numbers. You literally can't multiply two phasors to get zero because it doesn't exist in the phasor domain.
   
   
2. TAP> there's no way of reliably getting back the other argument
   
   The core of this issue is the existence (or not) of a reliable unbinding value for every possible unbinding value. However, let's assume you are setting up a VSA system where the domain of element values is essentially modular (i.e. discrete, evenly spaced integers), but for crazy engineering/implementation reasons the values are not evenly spaced on the number line (e.g. randomly displaced from the ideal position or the occasional ideal value is missing) I strongly suspect that such a system would still perform tolerably well as long as the inverses were (sufficiently) *approximately* correct. (For people who are aiming for neuromorphic implementations any technique that only works if it is exactly correct is not the technique for you.)

Cheers,

Ross

[Paxon Frady]

Hey guys,

I'll chime in here a bit. 

The sparse binding paper addresses some of these ideas. Here, we define sparse block code, originally described by Laiho. Like the phasors, and in the discussion of modular arithmetic, the binding operation can be related to the modulus sum. In this case the indices of the hot elements of the vector for each block are added modulo the block size. We also walk through the dollar of mexico example with the sparse block code (you need to use inverses as they are not self-inverses), and we show how the noise of the readout can be predicted by the VSA capacity theory. 

One thing to note about modular arithmetic in regards to VSA binding is also described in the VFA paper. For the complex FHRR vectors, if you choose the phases from a discrete set of the roots of unity (instead of continuous uniform distribution), then the encoding will "loop" or form a circular topology. This will happen with a modulo system or with the block code. This is because v^{k} = v^{k+M}, where M is the modulus base/size of blocks.

There can also be "harmonics" where similarity will increase above noise in certain situations, especially if the modulo size is not prime. This may also be something to watch out for with modulo sizes that are powers of 2.  

Tony:  Yes, in group theory/modular arithmetic there are "0 divisors," which would lead to the ambiguity you described. It would be interesting to consider an operation that multiplies phases and what that might mean. 

Paxon

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[George Owell]

Some observations on bundling operations:

In BSC, element-wise ties occur only with an equal number of 1s and
0s; bundling an odd number of elements cannot tie.
In MCR, ties are common because discrete phases can cancel in many ways.
In FHRR, exact phase cancellation is rare due to high precision elements.
In MAP (ternary), ties have an explicit representation as 0s. Ties
in the prior 3 frameworks are implicit.

Ties while bundling introduce noise. Could this be influencing your
experiment results?