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This notebook documents the implementation of the linear interpolation spline scalar encoder/decoder.
The reasoning behind the design choices is explained in XXX.
The encoder will map each unique scalar input value to a VSA vector such that similar input values are mapped to similar output VSA vectors.
For programming purposes, the mapping is represented by a spline_spec
object, which is created by vsa_mk_scalar_encoder_spline_spec().
The encoder specification represents a piecewise linear function from the input scalar value to another scalar value.
The piecewise linear function has \(k\) knots, which must be unique scalar values and given in increasing order.
Values of the input scalar that are outside the range of the knots are treated identically to the nearest extreme value of the knots.
There is a unique atomic VSA vector associated with each knot.
If the input scalar is exactly equal to a knot value then the encoder will return the corresponding VSA vector.
If the input scalar lies between two knot value then the encoder will return the weighted sum of the two corresponding VSA vectors with the weighting reflecting the position of the scalar value relative to the two knot values..
The piecewise linear function is specified by the knots given as an
argument to vsa_mk_scalar_encoder_spline_spec() and the VSA vectors
corresponding to the knots are randomly generated. The spline_spec
object captures these two components, which remain constant over the
simulation.
# function to make the specification for a piecewise linear spline encoder
vsa_mk_scalar_encoder_spline_spec <- function(
vsa_dim, # integer - dimensionality of VSA vectors
knots, # numeric vector - scalar knot locations (in increasing order)
seed = NULL # integer - seed for random number generator
) # value # data structure representing linear spline encoder specification
{
### Set up the arguments ###
# The OCD error checking is probably more useful as documentation
if(missing(vsa_dim))
stop("vsa_dim must be specified")
if(!(is.vector(vsa_dim, mode = "integer") && length(vsa_dim) == 1))
stop("vsa_dim must be an integer")
if(vsa_dim < 1)
stop("vsa_dim must be (much) greater than zero")
# if(!is.vector(sample_wt, mode = "numeric"))
# stop("sample_wt must be a numeric vector")
#
# if(length(sample_wt) < 2)
# stop("length(sample_wt) (the number of VSA vectors to be added) must be >= 2")
#
# if(min(sample_wt) < 0)
# stop("all weights must be >= 0")
#
# if(max(sample_wt) <= 0)
# stop("at least one weight must be > 0")
#
# # check that the specified seed is an integer
# if(!is.null(seed) && !(is.vector(seed, mode = "integer") && length(seed) == 1))
# stop("seed must be an integer")
# if seed is set the sampling specification vector is fixed
# otherwise it is randomised
# set.seed(seed)
# generate VSA atoms corresponding to each of the knots
tibble::tibble(
knots_scalar = knots,
knots_vsa = purrr::map(knots, ~ vsa_mk_atom_bipolar(vsa_dim = vsa_dim))
)
}Do some very small scale testing.
Generate a tiny spline_spec object and dsiplay the contents.
ss <- vsa_mk_scalar_encoder_spline_spec(vsa_dim = 10L, knots = c(-1, 1, 2))
ss# A tibble: 3 × 2
knots_scalar knots_vsa
<dbl> <list>
1 -1 <int [10]>
2 1 <int [10]>
3 2 <int [10]>ss$knots_vsa[[1]] [1] 1 -1 1 -1 -1 -1 -1 1 -1 1ss$knots_vsa[[2]] [1] -1 -1 1 -1 1 1 1 -1 -1 -1ss$knots_vsa[[3]] [1] 1 -1 1 1 1 1 -1 -1 1 1# function to encode a scalar numeric value to a VSA vector
# This function uses a linear interpolation spline
# to interpolate between a sequence of VSA vectors corresponding to the spline knots
vsa_encode_scalar_spline <- function(
x, # numeric[1] - scalar value to be encoded
spline_spec # data frame - spline spec created by vsa_mk_scalar_encoder_spline_spec()
) # numeric # one VSA vector, the encoding of the scalar value
{
### Set up the arguments ###
# The OCD error checking is probably more useful as documentation
if (missing(x))
stop("x must be specified")
if (!(is.vector(x, mode = "numeric") && length(x) == 1))
stop("x must be a numeric scalar")
if (is.na(x))
stop("x must be non-missing")
if (!is.finite(x))
stop("x must be finite")
if (missing(spline_spec))
stop("spline_spec must be specified")
if (
!(
is_tibble(spline_spec) &&
all(c("knots_scalar", "knots_vsa") %in% names(spline_spec))
)
)
stop("spline_spec must be a spline specification object")
# Map the scalar into a continuous index across the knots
# Linearly interpolate the input scalar onto a scale in which knots correspond to 1:n
i <- approx(
x = spline_spec$knots_scalar, y = seq_along(spline_spec$knots_scalar),
rule = 2, # clip x to fit the range of the knots
xout = x
)$y # get the interpolated value only
# Get the knot indices immediately above and below the index value
i_lo <- floor(i)
i_hi <- ceiling(i)
# Return the VSA vector corresponding to the index value
if (i_lo == i_hi) # check if index is on a knot
# Exactly on a knot so return the corresponding knot VSA vector
spline_spec$knots_vsa[[i]]
else {
# Between two knots
# Return the weighted sum of the corresponding knot VSA vectors
i_offset <- i - i_lo
vsa_add(
spline_spec$knots_vsa[[i_lo]], spline_spec$knots_vsa[[i_hi]],
sample_wt = c(1 - i_offset, i_offset)
)
}
}Do some very small scale testing.
Test what happens when the input scalar lies exactly on a knot.
vsa_encode_scalar_spline(-1.0, ss) [1] 1 -1 1 -1 -1 -1 -1 1 -1 1vsa_encode_scalar_spline( 1.0, ss) [1] -1 -1 1 -1 1 1 1 -1 -1 -1vsa_encode_scalar_spline( 2.0, ss) [1] 1 -1 1 1 1 1 -1 -1 1 1Test what happens when the input scalar falls outside the range of the knots.
vsa_encode_scalar_spline(-1.1, ss) [1] 1 -1 1 -1 -1 -1 -1 1 -1 1vsa_encode_scalar_spline( 2.1, ss) [1] 1 -1 1 1 1 1 -1 -1 1 1Check that intermediate values are random (becauase of the random
sampling in vsa_add()).
# remind us of the knot values
ss$knots_vsa[[1]] [1] 1 -1 1 -1 -1 -1 -1 1 -1 1ss$knots_vsa[[2]] [1] -1 -1 1 -1 1 1 1 -1 -1 -1# identify which elements are identical for the two knots
ss$knots_vsa[[1]] == ss$knots_vsa[[2]] [1] FALSE TRUE TRUE TRUE FALSE FALSE FALSE FALSE TRUE FALSE# interpolate midway between those two knots
vsa_encode_scalar_spline(0, ss) [1] 1 -1 1 -1 1 1 -1 -1 -1 1vsa_encode_scalar_spline(0, ss) [1] -1 -1 1 -1 1 1 1 -1 -1 -1vsa_encode_scalar_spline(0, ss) [1] 1 -1 1 -1 1 1 1 -1 -1 -1vsa_encode_scalar_spline(0, ss) [1] -1 -1 1 -1 1 -1 1 1 -1 -1Elements 3, 4, 6, 7, and 8 of the first and second knot vectors are identical, so the result of adding them is constant.
The other element values vary between the two knot vectors, so the
corresponding interpolated values will vary because of the random
sampling in vsa_add().
sessionInfo()R version 4.1.1 (2021-08-10)
Platform: x86_64-pc-linux-gnu (64-bit)
Running under: Ubuntu 21.04
Matrix products: default
BLAS: /usr/lib/x86_64-linux-gnu/blas/libblas.so.3.9.0
LAPACK: /usr/lib/x86_64-linux-gnu/lapack/liblapack.so.3.9.0
locale:
[1] LC_CTYPE=en_AU.UTF-8 LC_NUMERIC=C
[3] LC_TIME=en_AU.UTF-8 LC_COLLATE=en_AU.UTF-8
[5] LC_MONETARY=en_AU.UTF-8 LC_MESSAGES=en_AU.UTF-8
[7] LC_PAPER=en_AU.UTF-8 LC_NAME=C
[9] LC_ADDRESS=C LC_TELEPHONE=C
[11] LC_MEASUREMENT=en_AU.UTF-8 LC_IDENTIFICATION=C
attached base packages:
[1] stats graphics grDevices datasets utils methods base
other attached packages:
[1] tibble_3.1.3 Matrix_1.3-4 purrr_0.3.4 dplyr_1.0.7
[5] DiagrammeR_1.0.6.1
loaded via a namespace (and not attached):
[1] Rcpp_1.0.7 pillar_1.6.2 compiler_4.1.1 later_1.2.0
[5] RColorBrewer_1.1-2 git2r_0.28.0 workflowr_1.6.2 tools_4.1.1
[9] digest_0.6.27 lattice_0.20-44 jsonlite_1.7.2 evaluate_0.14
[13] lifecycle_1.0.0 pkgconfig_2.0.3 rlang_0.4.11 rstudioapi_0.13
[17] cli_3.0.1 yaml_2.2.1 xfun_0.25 stringr_1.4.0
[21] knitr_1.33 generics_0.1.0 fs_1.5.0 vctrs_0.3.8
[25] htmlwidgets_1.5.3 grid_4.1.1 tidyselect_1.1.1 rprojroot_2.0.2
[29] glue_1.4.2 here_1.0.1 R6_2.5.0 fansi_0.5.0
[33] rmarkdown_2.10 bookdown_0.22 magrittr_2.0.1 whisker_0.4
[37] promises_1.2.0.1 ellipsis_0.3.2 htmltools_0.5.1.1 renv_0.14.0
[41] httpuv_1.6.1 utf8_1.2.2 stringi_1.7.3 visNetwork_2.0.9
[45] crayon_1.4.1