Category Archives: data analysis

A Critique of The Programming Language J

I’ve spent around a year now fiddling with and eventually doing real
data analytic work in the The Programming Language J. J is one of
those languages which produces a special enthusiasm from its users and
in this way it is similar to other unusual programming languages like
Forth or Lisp. My peculiar interest in the language was due to no
longer having access to a Matlab license, wanting an array oriented
language to do analysis in, and an attraction to brevity and the point
free programming style, two aspects of programming which J emphasizes.

Sorry, Ken.

Sorry, Ken.

I’ve been moderately happy with it, but after about a year of light
work in the language and then a month of work-in-earnest (writing
interfaces to gnuplot and hive and doing Bayesian inference and
spectral clustering) I now feel I am in a good position to offer a
friendly critique of the language.

First, The Good

J is terse to nearly the point of obscurity. While terseness is not a
particularly valuable property in a general purpose programming
language (that is, one meant for Software Engineering), there is a
case to be made for it in a data analytical language. Much of my work
involves interactive exploration of the structure of data and for that sort
of workflow, being able to quickly try a few different ways of
chopping, slicing or reducing some big pile of data is pretty
handy. That you can also just copy and paste these snippets into some
analysis pipeline in a file somewhere is also nice. In other words,
terseness allows an agile sort of development style.

Much of this terseness is enabled by built in support for tacit
programming. What this means is that certain expressions in J are
interpreted at function level. That is, they denote, given a set of
verbs in a particular arrangement, a new verb, without ever explicitly
mentioning values.

For example, we might want a function which adds up all the maximum
values selected from the rows of an array. In J:


J takes considerable experience to read, particularly in Tacit
style. The above denotes, from RIGHT to LEFT: for each row ("1)
reduce (/) that row using the maximum operation >. and then (@:)
reduce (/) the result using addition (+). In english, this means:
find the max of each row and sum the results.

Note that the meaning of this expression is itself a verb, that is
something which operates on data. We may capture that meaning:

sumMax =: +/@:(>./"1)

Or use it directly:

+/@:(>./"1) ? (10 10 $ 10)

Tacit programming is enabled by a few syntactic rules (the so-called
hooks and forks) and by a bunch of function level operators called
adverbs and conjuctions. (For instance, @: is a conjunction rougly
denoting function composition while the expression +/ % # is a fork,
denoting the average operation. The forkness is that it is three
expressions denoting verbs separated by spaces.

The details obscure the value: its nice to program at function level
and it is nice to have a terse denotation of common operations.

J has one other really nice trick up its sleeve called verb
. Rank itself is not an unusual idea in data analytic languages:
it just refers to the length of the shape of the matrix; that is, its

We might want to say a bit about J’s basic evaluation strategy before
explaining rank, since it makes the origin of the idea more clear. All
verbs in J take one or two arguments on the left and the right. Single
argument verbs are called monads, two argument verbs are called dyads.
Verbs can be either monadic or dyadic in which case we call the
invocation itself monadic or dyadic. Most of J’s built-in operators
are both monadic and dyadic, and often the two meanings are unrelated.

NB. monadic and dyadic invocations of <
4 < 3 NB. evaluates to 0
<3 NB. evalutes to 3, but in a box.

Give that the arguments (usually called x and y respectively) are
often matrices it is natural to think of a verb as some sort of matrix
operator, in which case it has, like any matrix operation, an expected
dimensionality on its two sides. This is sort of what verb rank is
like in J: the verb itself carries along some information about how
its logic operates on its operands. For instance, the built-in verb
-: (called match) compares two things structurally. Naturally, it
applies to its operands as a whole. But we might want to compare two
lists of objects via match, resulting in a list of results. We can
do that by modifying the rank of -:

x -:”(1 1) y

The expression -:”(1 1) denotes a version of match which applies to
the elements of x and y, each treated as a list. Rank in J is roughly
analogous the the use of repmat, permute and reshape in Matlab: we can
use rank annotations to quickly describe how verbs operate on their
operands in hopes of pushing looping down into the C engine, where
it can be executed quickly.

To recap: array orientation, terseness, tacit programming and rank are
the really nice parts of the language.

The Bad and the Ugly

As a programming environment J can be productive and efficient, but it
is not without flaws. Most of these have to do with irregularities in
the syntax and semantics which make the language confusing without
offering additional power. These unusual design choices are
particularly apparent when J is compared to more modern programming

Fixed Verb Arities

As indicated above, J verbs, the nearest cousin to functions or
procedures from other programming languages, have arity 1 or
arity 2. A single symbol may denote expressions of both arity, in
which case context determines which function body is executed.

There are two issues here, at least. The first is that we often want
functions of more than two arguments. In J the approach is to pass
boxed arrays to the verb. There is some syntactic sugar to support
this strategy:

multiArgVerb =: monad define
‘arg1 arg2 arg3’ =. y
NB. do stuff

If a string appears as the left operand of the =. operator, then
simple destructuring occurs. Boxed items are unboxed by this
operation, so we typically see invocations like:

multiArgVerb('a string';10;'another string')

But note that the expression on the right (starting with the open
parentheses) just denotes a boxed array.

This solution is fine, but it does short-circuit J’s notion of verb
rank: we may specify the the rank with which the function operates on
its left or right operand as a whole, but not on the individual
“arguments” of a boxed array. But nothing about the concept of rank
demands that it be restricted to one or two argument functions: rank
entirely relates to how arguments are extracted from array valued
primitive arguments and dealt to the verb body. This idea can be
generalized to functions of arbitrary argument count.

Apart from this, there is the minor gripe that denoting such single
use boxed arrays with ; feels clumsy. Call that the Lisper’s bias:
the best separator is the space character.1

A second, related problem is that you can’t have a
zero argument function either. This isn’t the only language where
this happens (Standard ML and OCaml also have this tradition, though I
think it is weird there too). The problem in J is that it would feel
natural to have such functions and to be able to mention them.

Consider the following definitions:

o1 =: 1&-
o2 =: -&1

(o1 (0 1 2 3 4)); (o2 (0 1 2 3 4))
│1 0 _1 _2 _3│_1 0 1 2 3│

So far so good. Apparently using the & conjunction (called “bond”)
we can partially apply a two-argument verb on either the left or the
right. It is natural to ask what would happen if we bonded twice.


Ok, so it produces a verb.

 3 3 $ ''
  ;((o1&1 (0 1 2 3 4))
  ; (o2&1 (0 1 2 3 4))
  ; (1&o1 (0 1 2 3 4))
  ; (1&o2 (0 1 2 3 4)))

│     │o1          │o2          │
│right│1 0 1 0 1   │1 0 _1 _2 _3│
│left │1 0 _1 _2 _3│_1 0 1 2 3  │

I would describe these results as goofy, if not entirely impossible to
understand (though I challenge the reader to do so). However, none of
them really seem right, in my opinion.

I would argue that one of two possibilities would make some sense.

  1. (1&-)&1 -> 0 (eg, 1-1)
  2. (1&-)&1 -> 0″_ (that is, the constant function returning 0)

That many of these combinations evaluate to o1 or o2 is doubly
confusing because it ignores a value AND because we can denote
constant functions (via the rank conjunction), as in the expression


What this is all about is that J doesn’t handle the idea of a
function very well. Instead of having a single, unified abstraction
representing operations on things, it has a variety of different ideas
that are function-like (verbs, conjuctions, adverbs, hooks, forks,
gerunds) which in a way puts it ahead of a lot of old-timey languages
like Java 7 without first order functions, but ultimately this
handful of disparate techniques fails to acheive the conceptual unity
of first order functions with lexical scope.

Furthermore, I suggest that nothing whatsoever would be lost (except
J‘s interesting historical development) by collapsing these ideas
into the more typical idea of closure capturing functions.

Other Warts

Weird Block Syntax

Getting top-level2 semantics right is hard in any
language. Scheme is famously ambiguous on the subject, but at
least for most practical purposes it is comprehensible. Top-level has
the same syntax and semantics as any other body of code in scheme
(with some restrictions about where define can be evaluated) but in
J neither is the same.

We may write block strings in J like so:

blockString =: 0 : 0 
Everything in here is a block string.       

When the evaluator reads 0:0 it switches to sucking up characters
into a string until it encounters a line with a ) as its first
character. The verb 0:3 does the same except the resulting string is
turned into a verb.

plus =: 3 : 0

However, we can’t nest this syntax, so we can’t define non-tacit
functions inside non-tacit functions. That is, this is illegal:

plus =: 3 : 0
  plusHelper =. 3 : 0
  x plusHelper y

This forces to the programmer to do a lot of lambda lifting
manually, which also forces them to bump into the restrictions on
function arity and their poor interaction with rank behavior, for if
we wish to capture parts of the private environment, we are forced to
pass those parts of the environment in as an argument, forcing us to
give up rank behavior or forcing us to jump up a level to verb


Of course, you can define local functions if you do it tacitly:

plus =: 3 : 0
    plusHelper =. +
    x plusHelper y   

But, even if you are defining a conjunction or an adverb, whence you
are able to “return” a verb, you can’t capture any local functions –
they disappear as soon as execution leaves the conjunction or adverb

That is because J is dynamically scoped, so any capture has to be
handled manually, using things like adverbs, conjunctions, or the good
old fashioned fix f., which inserts values from the current scope
directly into the representation of a function. Essentially all modern
languages use lexical scope, which is basically a rule which says: the
value of a variable is exactly what it looks like from reading the
program. Dynamic scope says: the valuable of the variable is whatever
its most recent binding is.


The straight dope, so to speak, is that J is great for a lot of
reasons (terseness, rank) but also a lot of irregular language
features (adverbs, conjunctions, hooks, forks, etc) which could be
folded all down into regular old functions without harming the
benefits of the language, and simplifying it enormously.

If you don’t believe that regular old first order functions with
lexical scope can get us where we need to go, check out my
tacit-programming libraries in R and Javascript. I
even wrote a complete, if ridiculously slow implementation of J‘s
rank feature, literate-style, here.


1 It bears noting that ; in an expression like (a;b;c)
is not a syntactic element, but a semantic one. That is, it is the
verb called “link” which has the effect of linking its arguments into
a boxed list. It is evaluated like this:


(a;b;c) is nice looking but a little strange: In an expression
(x;y) the effect depends on y is boxed already or not: x is always boxed regardless, but y is boxed only if it wasn’t boxed before.

2 Top level? Top-level is the context where everything
“happens,” if anything happens at all. Tricky things about top-level
are like: can functions refer to functions which are not yet defined,
if you read a program from top to bottom? What about values? Can you
redefine functions, and if so, how do the semantics work? Do functions
which call the redefined function change their behavior, or do they
continue to refer to the old version? What if the calling interface
changes? Can you check types if you imagine that functions might be
redefined at any time? If your language has classes, what about
instances created before a change in the class definition. Believe or
not, Common Lisp tries to let you do this – and its confusing!

On the opposite end of the spectrum are really static languages like
Haskell, wherein type enforcement and purity ensure that the top-level
is only meaningful as a monolith, for the most part.

In defense of “The Thin Line Aesthetic”

I was lucky to be one of the guest artists at the Code+Art Student Visualization contest at NCSU library recently, where parts of my generative art work Clocks were displayed. In preparation for the show, I wrote specific pieces for the space, which uses a large screen made out of Christie Micro-tiles. These are modular screens which can be used to construct large and even irregular displays. While the Micro-tiles gave me the largest amount of screen real-estate I’d ever worked with, they posed their own challenges. In particular, the luminance tends to vary between tiles in such a way that art works with predominantly white backgrounds can be distracting, since each tile making up the whole screen will appear at a different brightness. The effect is less noticeable when darker colors are supplied.

Thin Lines!

Thin Lines!

During these discussions, it was suggested that using dark backgrounds might help get us away from “the thin line aesthetic” so predominant in generative art. I agree, thin lines, typically dark on white, are extremely common in generative artworks. But here I will say a few words in their defense.

Clock 10 is a thin-line clock. I’ve had occasion to think very carefully about why this clock works as a generative art piece, and it is typically the clock I talk about when talking about the project as a whole because it has a relatively simple, but non-trivial, account. Briefly, Clock 10’s charged particles want to distribute themselves evenly over the face of the clock (since they are all positively charged, and hence repel one another). The clock hands persistently frustrate this tendency by moving particles from the second hand to the hour or minute hand. As such, the particles are constantly seeking, but never attaining, their low energy equilibrium state. Critically, there is not just one such equilibrium state: there exists a family of states related by symmetry transforms (continuous and discrete rotations).

What this boils down to is that Clock 10 traces out the symmetries of the ground states. This is why, if you let the clock run for a half hour, you see concentric rings appear: these rings are the places that particles would like to be modulo rotations, were they allowed to find those states without interference.


A few aesthetic choices support the relationship of Clock 10 to this interpretation. The color of the trace left by each particle is adjusted in such a way that it only darkens as the particle settles down, so that paths near equilibrium are dark while those far from it are white. And, of course, we use thin lines, which allow lots of information about those trajectories to appear on the face of the clock.

I am a barbarian, so far as artistic pedigree is concerned, but if Clock 10 lives anywhere in the landscape of the practice of generative art, it is in the school of Complexism. Complexism suggests one role of generative art is to explore complex systems. In the sense the Clock 10 is an aesthetically pleasing visualization of the ground states of a certain physical system, it meets this criterion. And thin lines help it operate in this way because they allow us to see a lot of different trajectories in a small amount of space. They give the clock non-trivial texture: tendencies of the motion can be apprehended at a large scale while details of the motion are still discernable.

I tried a variety of other ways of visualizing the trajectories, but none were particularly satisfying because they obscured the fine-scale variations in a way which significantly reduced the information content of the visualization. Part of the impact of generative art is that it imitates nature, to an extent, in that it can compound over and over again many fine motions. The accumulation of so many effects is part of the immediate perception of a work, and undermining it undermines one of the fundamental advantages of using computers, systems capable simultaneously of great precision and great, repetitive patience.

So use thin lines! Or, if you are seeking alternative aesthetic choices, try to find ones which capture the same benefits, packing lots of precise detail into the image in such a way that larger trends are also made visible.


Aping J’s Verb Rank in Puff

This blog post will sketch out some thoughts relating to Puff, a function level programming language I am embedding in Javascript and J‘s notion of operator rank.

Rank as it pertains to nouns in J is fairly easy to understand: it is just the number of dimensions of an array. Scalars, like 10, have rank 0, the empty value (denotable as 0 $ 0) has rank 0, a simple vector (0 0 0) has rank 1, a 2d matrix rank 2, etc.

But J also has rank for verbs. Consider the verb +.

(1 2 3) + (4 5 6)
-> 5 7 9

(For J tyros: + is called a verb in J and furthermore we use it in its dyadic sense, which is to say we pass it arguments on the left and the right.)

Informally we understand from this that in J + operates element-wise on both its left and right operands. This means its left and right rank are both zero and it operates, then, on the rank zero elements of its arguments: the individual scalar values.

But there is more to the story. As a side note, We can denote multi-dimensional arrays in J like so:

]example =. 2 3 $ 1 2 3 4 5 6 
1 2 3
4 5 6

(For the curious, that is “change the shape ($) for array 1 2 3 4 5 6 so that it is 2 3)

J has a box operator which is handy for demonstrating rank issues. It is denoted < and wraps any value into a box, which is a special scalar value type which holds something else.


Operators in J have rank and the rank of < is infinite. This means that it always operates on its argument as a whole.

<(1 2 3 4 5 6)
│1 2 3 4 5 6│

But the smart thing about J is that you can modify verbs with adverbs one of which returns a new verb with a different rank. See if you can guess what all this means:

<"0(1 2 3 4 5 6)

The array denotation 1 2 3 4 5 6 is the same as before, but now we have written <"0 instead of <. " is an adverb which modified its right hand arguments’ rank such that it is the left hand value. The result of<"0 then is a verb with the same meaning as < except that it has rank 0. Verbs with rank 0 operate on the smallest possible cells of the array, so that

<"0(3 2 $ 1 2 3 4 5 6)

each element of the input is boxed regardless of the incoming arrays shape or rank.

If we use a different rank:

<"1(3 2 $ 1 2 3 4 5 6)
│1 2│3 4│5 6│

We get a different result. One-ranked verbs operate 1-cells (that is, the elements of rank 1) of the incoming array, in this case the arrays 1 2, 3 4, and 5 6.

The rules work for dyadic values too – each argument of the verb has a rank (a right rank and a left rank) which determines how the underlying logic represented by the verb is used to join elements from the right and left arguments.

By modifying verb rank you can custom tailor your iteration through argument arrays and avoid most explicit looping.


Puff is mostly aping the function level semantics of J but we can analogize verb rank too. Consider the Puff function map, which has a single argument meaning:

var plus1 = _p(plus,1);
map(plus1)([1,2,3]) -> [2,3,4]

plus1 above would have, in J an infinite rank: it always applies to its whole argument. When we say map(plus1) we have a new function which applies to the N-cells of its argument (in this case integers). In other words, map creates a new function which peels off one layer of its input and applies the original function, collecting the outputs.

What, then, is

var mm_plus1 = map(map(plus1))


(NB, we can denote this in Puff via rep(map,2,plus1))

Here is a hint:

mm_plus1([[1,2,3],[4,5,6]]) -> [[2,3,4],[5,6,7]]

Now we have a function operating on the N-2 cells of the input. Rank in J typically operates bottom up: we start at rank 0 operating on the 0 cells, and increasing rank operates on larger and larger chunks of the input array. In contrast, iterative application of map in Puff moves us from the larger chunks to smaller and smaller chunks, until a number of applications equal to the array rank has us operating on individual items.

J being J we can mimic this behavior using negative rank.

<"_2(3 2 $ 1 2 3 4 5 6)

(_2 denotes the number -2 in J for possibly obscure reasons to do with making the parser simpler.)

Given that 3 2 $ 1 2 3 4 5 6 has rank 2, the verb <"_2 must operate on the 2-2=0 cells.

The J approach of, by default, thinking about rank from 0-cells up works well for that language because matrices in J are regular and they keep track of their rank. If we represent matrices as nested arrays in Javascript (this is not the only option, but it is the most idiomatic) then the real rank of a matrix cannot be known without a full traversal, which is prohibitive.

I might, one day, integrate a multidimensional matrix implementation into Puff and then enable rank modifying functions to work on that representation, but for now I want to focus on the successive use ofmap to simulate ranking down a function from infinite rank.

Consider Rank

Consider the following definition:

function rankedCall(f,n,a){
        return rep(map, n, f)(a);
    } else {
        throw new Error("Positive ranks not yet supported.");

var rank = _c(rankedCall);

Such that:

rank(plusOne,1)([1,2,3]) -> [2,3,4]

Cute. This gets us somewhere. But what really makes rank useful is that each argument carries its own rank and the system resolves the looping for you. In J operators have at most two arguments (to which rank applies, simulating more arguments with lists of boxes bypasses ranked dispatch).

Dealing with multiple argument functions is tricky. Let’s start with two.


// Puff provides a plus function
plus(1,3) -> 4
// but it doesn't work on arrays
plus([1,2,3],[4,5,6]) -> '1,2,34,5,6'

That last line is because Javascript idiotically interprets [1,2,3]+[4,5,6] to mean [1,2,3].toString()+[4,5,6].toString().

For these one dimensional arrays, we can get what we want with map which applies a function f of arity n to the items of n arrays.

map(plus,[1,2,3],[4,5,6]) -> [5,7,9]

(NB. In Puff we can also have said map(plus)([1,2,3],[4,5,6]))

What if we have [1,2,3] and [[4],[5],[6]], that is, the second argument is rank two?

Put aside questions of efficiency for a moment and consider the function:

function nextCellIndex(a, indexes){
    var indexes =; // copy the array
    var delta = indexes.length-1;
    var subIndex = indexes.slice(0,delta);
    var indexedArray = index.apply(null, [a].concat(subIndex));
    var done = indexes[delta]+1 < indexedArray.length;
      delta = delta -1;
          return null;
      } else {
          indexedArray = index.apply(null, [a].concat(indexes.slice(0,delta)));
          done = indexes[delta]+1 < indexedArray.length;

    indexes[delta] = indexes[delta]+1;
    for(var i = delta+1; i<indexes.length; i = i + 1){
      indexes[i] = 0;
    return indexes;

This function takes an array and an array of indexes and finds the next valid index into that array by incrementing the innermost index, checking whether that is in bounds, stopping if it is, or incrementing the next innermost and so on. If there is no valid next index, then null is returned.

If we want what J would call the -2 cells of an array a, we iteratively apply this function to a two element index vector.

var a = [[1],[2,3],[4]]
var indexes = repeatAccumulate(_p(nextCellIndex,a),3,[0,0])

Evaluating to:

[ [ 0, 0 ], [ 1, 0 ], [ 1, 1 ], [ 2, 0 ] ]

that is, the indexes of the -2 cells. We can get these by, for instance,

index.apply(null, a, indexes[0])

Note that a is not a regular matrix (the second item of a has a different length than the first and third – it has no obvious rank, but we can talk about its n-cells if we talk about them from outside in. We can write a function to give us these cells:

function cells(n, a){
      var nn = -n;
      var out = [];
      var indexes = initArray(nn,0);
          out.push(index.apply(null, [a].concat(indexes)));
          indexes = nextCellIndex(a,indexes)
      return out;
    } else if (n===0){
      return a;
    } else {
      throw new Error("Positive cells not yet supported.");

We can then just fall back onto map with the appropriate applications of cells:

-> [ 2, 4, 6 ]

Conceptually we’ve done well for ourselves: we’ve reproduced J‘s ability to change the way that functions join elements of arrays of arbitrary dimension. On top of that, by virtue of the arity of map, which can apply a function of any arity to any number of arrays, we have extended this idea to operators of any number of arguments (J is limited to monadic and dyadic verbs.)

In addition, Puff allows us to write the above function in a point free fashion:

var ex = f(2,au(map, al(plus), n0, r(n1,_p(cells, 2))));
-> [2, 4, 6]

(NB. al returns a function which always returns the argument to al, short for always, n0 returns the first element of a thing, n1 the second, etc. f (short for lambda) returns a function which first collects its arguments into an array and then passes them through its subsequent arguments as if via r (rCompose). Finally, au (short for augment) takes a set of functions and returns a new function which transforms its inputs via functions 1..n and applies function 0 to the that list.)

Positive Ranks

Using negative ranks is much more in-line with idiomatic javascript, since there are no native multidimensional arrays. We can produce a simple implementation of positive ranks if we make a few simple assumptions about usage. First consider:

function guessRank(a){
    var rank = 0;
    var done = false;
        if(typeof a['length'] === 'number'){
          rank = rank + 1;
          a = a[0];
        } else {
          done = true;
    return rank;

Working like:

:Puff:> guessRank(10)
:Puff:> guessRank([1,2,3])
:Puff:> guessRank([[1,2],[2,3],[3,4]])

The assumption we are making here is that the rank of sub-elements is homogeneous (and hence, the first element is a sufficient indicator). Now that we can guess the rank of an array, we can fill in the positive rank branch of our cells function:

function cells(n, a){
      var nn = -n;
      var out = [];
      var indexes = initArray(nn,0);
          out.push(index.apply(null, [a].concat(indexes)));
          indexes = nextCellIndex(a,indexes)
      return out;
    } else {
      var rank = n-guessRank(a);
      return cells(rank, a);

Now we can finally write our implementation of J‘s conjunction ". Our version of " will be called rank and will take a function and a list of ranks and return a new function with the appropriate rank behavior.

function rank(f){
    var ranks =,1,arguments.length);
    return function(){
    return map.apply(null,[f].concat(map(cells, 

We can now say:


And get [5,7,9]. Just like J. Of course, as we’ve written the code here we won’t be anywhere near the efficiency of J – in particular we iterate over each argument array separately, where we could combine all those loops into just one. But performance isn’t everything and we can always optimize the Puff implementation as needed. Rewriting the approprite sequence functions (map,mapcat,crossMap) to handle lazy versions of the sequences and introducing a lazy cells operator would be the most elegant solution. I’m sure I’ll get there eventually.

In the meantime, I hope I’ve at least helped the reader understand J‘s rank concept in greater depth and also showed off some of the nice ways Puff can simulate J style while staying entirely in Javascript.

Spectral Clustering in J

With Examples from Quantitative Neuroscience

In my never ending quest to understand hard, but useful stuff, I’ve been learning the Programming Language J. J is an array oriented language in the APL family. Matlab is a related language, but its APL-ness is quite attenuated compared to J. I know Matlab already, having used it quote extensively as a research scientist, so J is not utterly alien, but unlike Matlab, which presents a recognizable syntactic face to the user, J retains an APL-like syntax.

As you will see, this presents challenges. The best way to learn to use a tool is to use it, so I thought I would take the opportunity to investigate a clustering technique which would have been useful to me as a grad student (had I known about it): Spectral Clustering.

Spectral Clustering1 refers to a family of clustering approaches which operate on the assumption that local information around data points is more useful for the purposes of assigning those points clusters than is their global arrangement. Typical clustering techniques, like k-means, make the assumption “points in a cluster will be near to eachother.” Spectral clustering relaxes that assumption to “points in a cluster will be near to other points in their cluster.”

We will develop the spectral clustering algorithm and then apply it to a series of more interesting data sets.

Part 1: 1d Gaussian Clusters

We are going to work on exploring spectral clustering. In order to do that we need something to cluster. Our first data set will be trivial: made by combining two sets drawn from different normal distributions2.

load 'utils.ijs'
load 'km.ijs'
require 'graphics/plot'
require 'numeric'
require 'viewmat'
require 'math/lapack'
require 'math/lapack/geev'

clusterSize1 =: 100
e1d1 =: (0 0.01) gausian clusterSize1
e1d2 =: (13 0.01) gausian clusterSize1

e1labeled =:  (((clusterSize1$0),(clusterSize1$1))) ,: (e1d1, e1d2)
e1data =: 1 { e1labeled
e1labels =: 0 { e1labeled

NB. show the histogram of the data
NB. (_6 6 200) histPlot e1data

The commented line produces the following plot:

Example One Data Histogram

This kind of data is trivial to cluster with an algorithm like k-means3.

kmClustering  =: 2 km (((# , 1:) e1data) $ e1data)

+/ (kmClustering = e1labels)

Should give ~ 200, unless something unusual happens.

As I suggested in the introductory section, the magic of spectral clustering is that it works on a representation of data which favors local information about the relationships between objects. Hence we will need to create some sort of connectivity graph for our data points. There are lots of approaches, but the upshot is that i,j in our connectivity matrix must tell us whether it is likely that point i and point j are in the same cluster.

For this simple example, we will just use the Euclidean distance between points and a threshold: anything lower than the threshold is connected and anything higher is not.

NB. d is the distance between two vectors of equal dimension
d =: %:@:+/@:*:@:([ - ])

NB. ds is the table of all pairwise distances where each argument is a
NB. list of vectors.
ds =: d"(1 1)/

NB. convert a list of points to a list of 1d vectors
l2v =: (#@:],1:) $ ]

NB. ds for lists of points
ds1d =: l2v@:] ds l2v@:]

NB. the table of all distances of x
dof =: ] ds ]

NB. the same except for a list of points
dof1d =: ] ds1d ] 

connections1d =: [ <~ dof1d@:]

e1connections =: 1.0 connections1d e1data

NB. viewmat e1connections

The connections matrix looks like:

Example 1 connections matrix

Spectral clustering operates on a Matrix called the Graph Laplacian which is related to the connectivity matrix. To get the laplacian matrix L we subtract the connectivity matrix from something called the degree matrix. Explained clearly but without interpretation, the degree matrix is just the sum of all connections for a given vertex in our graph, placed on the diagonal.

calcDegreeMat =:  (eye@:#) * +/"1 

The degree matrix just tells you how connected a given vertex is. To get the Laplacian Matrix you subtract the connectivity matrix from the degree matrix. This is conveniently defined in J:

calcLap =: calcDegreeMat - ]

For our first data set, this Laplacian looks like:

Example 1 Laplacian Matrix

Brief meditation will explain why the components of this matrix representing inter-object connections are so faint compared to those on the diagonal: the rank of each vertex in the graph is always larger than its connections, being the sum thereof.

Now that we have the Laplacian I will perform:

Spectral Clustering

The paper A Tutorial Introduction to Spectral Clustering, describes a family of spectral clustering strategies. We will implement just the first.

It involves the steps:

  1. Find the graph Laplacian for your system
  2. Find its eigenvalues and eigenvectors
  3. Sort the latter by the former, in ascending order
  4. Arrange the vectors as columns of a matrix
  5. truncate that matrix to a small number of columns, corresponding to the smaller eigenvalues.
  6. treat the rows of the resulting matrix as vectors.
  7. perform clustering, say k-means, on those vectors.

We can denote these succinctly in J:

eigensystem =: monad define 
  raw =. geev_jlapack_ y
  ii =. /:@:>@:(1&{) raw
  lvecs =. |: (ii { |: (0 unboxFrom raw))
  vals =. ii { (1 unboxFrom raw)
  rvecs =. |: (ii { :| (2 unboxFrom raw))
  lvecs ; vals ; rvecs  

eigenvectors =: >@:(0&{@:eigensystem)
eigenvalues =: >@:(1&{@:eigensystem)

takeEigenvectors =: |:@([ {. |:@:])

e1clustering =: 2 km takeEigenvectors eigenvectors e1Lap

(+/ e1clustering = e1labels) NB. should be about 200

Example 1 Clustered

A boring, if resounding, success.

Example 2: Concentric Clusters

Consider the following data:

rTheta =: (([ * cos@:]) , ([ * sin@:]) )"(0 0)
e2clusterSize =: 100
e2d1 =: ((1,0.1) gausian e2clusterSize) rTheta ((0,2p1) uniform     e2clusterSize)
e2d2 =: ((4,0.1) gausian (2*e2clusterSize)) rTheta ((0,2p1) uniform     (2*e2clusterSize))
e2data =: e2d1, e2d2

Concentric Clusters

This is the sort of data set which is pathological for clustering procedures which work in the original coordinates of the data. Even traditional dimension reducing techniques like principal component analysis will fail to improve the performance (in fact, the principal components of this data set are analogous to x and y, modulo a rotation).

We could, of course, realize by inspection that this problem reduces to the first example if the angular component is thrown away and we cluster on the radial component only.

However, in general, we are working with data in n dimensions and such simplifications may not be apparent to us. Hence, we will proceed with this example set.

Again, we produce the connectivity matrix by simple ci,j = di,j < epsilon.

e2connections =: 1 connections e2data 
e2Lap =: calcLap e2connections
e2ids =:  2 km 2 takeEigenvectors eigenvectors e2Lap

e2 connectivity matrix

As we physicists like to say: now we just turn the crank:

e2Lap =: calcLap e2connections
e2ids =:  2 km 2 takeEigenvectors eigenvectors e2Lap

And we can plot these points labeled by ID:

e2 data clustered

Pretty neat - we have successfully clustered a data set which traditional techniques could not have dealt without, and without manually figuring out a transform the simplified the data set.

Example 3: Neural Spike Trains

My interest in spectral clustering goes back to my graduate research, which involved, among other things, the automatic clustering of neural data. Neurons, in animals like us, anyway, communicate with one another using brief, large depolarizations of the voltage across their cell membranes. Neuroscientists call these events spikes and they literally look like large (~80 mV) spikes of voltage in the recording of the voltage across a cell membrane. These spikes travel down the neuron's axon and are transmitted through synapses to other neurons. There are other forms of communication between neurons, but it is generally assumed that spikes underly much of the computations that neural networks perform.

Spikes are short (1-2 ms) and firing rates are generally low on this time scale, with a typical inter-spike period of 10-100 ms (depending a great deal on the inputs to the neuron). In order to understand how neurons work, neuroscientist repeat a stimulus over and over again and record the spike times relative to the beginning of the experiment.

This results in a data set like:

33.0  55.2  80.0
30.0  81.1
55.2  85.9

Where rows are trials and each number is a spike time for that trial. In J we can read this data as padded arrays:

spikes =: asArray 0 : 0
33.0  55.2  80.0
30.0  81.1
55.2  85.9

And J will pad it out with zeros.

 33   55.2   80
 30   81.1   0
 55.2 85.9   0

Under certain circumstances, during such experiments, neurons generate a small number of distinct spiking patterns. Assuming that these patterns may be of interest, much of my research focused on extracting them automatically. That is: given a data set of spike trains, extract the clusters of similar trains.

Unfortunately, spike trains are not vectors and so there is no convenient euclidean distance measure one can easily apply to them. Luckily Victor and Purpura (1996) developed an edit-distance metric on spike trains by analogy to similar metrics other, non-vector, data (like gene sequences).

The Victor-Purpura Metric defines the distance between two spike trains as the minimum cost of distorting one spike train into the other using just two operations: sliding spikes in time to line up or deleting/adding spikes. The cost of sliding spikes in time is has the parameterized cost q*dt where dt is the time difference, and the cost of adding or removing a spike is one.

The algorithm to find the minimum such distortion can be easily implemented:

vpNaive =: adverb define 
  'q' =. m
  's1' =. nonzeros x 
  's2' =. nonzeros y
  'n1' =. #s1
  'n2' =. #s2 
  if. n1 = 0 do.
  elseif. n2 = 0 do.
  elseif. 1 do.
    'h1' =. {. s1
    'h2' =. {. s2
    'movecost' =. q * abs (h1-h2)
    if. movecost < 1.0 do.
      'restcost' =. ((}. s1) (q vpNaive) (}. s2))
      restcost + movecost 
      'restcost' =. ((}. s1) (q vpNaive) (}. s2))
      restcost + 1

although much more efficient implementations are possible. For our purposes, this will suffice.

With this definition, we can produce the similarity matrix for a given q using this J adverb:

spikeDistances =: adverb define   
  y ((m vpNaive)"(1 1) /) y

The choice of q is an important issue which we will neglect for now. See my doctoral thesis for insight into this question. The one sentence upshot is that q should generally be chosen to maximize the information content in the resulting similarity matrix.

Now let's create an example data set consisting of two distinct firing patterns. A firing pattern consists of a mean time for each "event" in the pattern, plus that events "reliability," which is the probability that a spike actually appears in an event. An event may be localized at a very specific time, but spikes may only sometimes appear on a given trial in that event.


NB. This produces a fake spike data set of Y trials with events
NB. specified by means, stds, and reliabilities, passed in as a
NB. rectangular matrix.
makeSpikes =: dyad define 
  'means stds reliabilities' =. x
  'n' =. y

  spikes =. (n,#means) $ means 
  spikes =. spikes + ((n,#stds) $ stds)*(rnorm (n,#stds))
  spikes =. spikes * ((runif (n,#reliabilities))<((n,#reliabilities) $   reliabilities))
  nonzeros"1 spikes

This function takes a description of the pattern, like:

e3pattern1 =: asArray 0 : 0 
  10 20 30
  1  1.5  1.5
  0.9 0.9 0.6

e3pattern2 =: asArray 0 : 0
  15 40
  1  1
  0.8 0.8

And returns a data set of N trials:

NB. Now we use makeSpikes to generate the actual data sets.
e3spikes1 =: e3pattern1 makeSpikes 50
e3spikes2 =: e3pattern2 makeSpikes 50

NB. and the utility `concatSpikes` to combine them.  This just makes
NB. sure the zero padding works.
e3spikes =: e3spikes1 concatSpikes e3spikes2

We can plot this data in the traditional fashion, as a "rastergram," where each spike is plotted with its time as the x-coordinate and its trial as its y-coordinate:

rastergram e3spikes

Example 3 Rastergram

It is interesting to note how difficult it can be to see these patterns by visual inspection if the trials are not sorted by pattern:

rastergram shuffle e3spikes

Example 3 Rastergram shuffled

Now let's cluster them:

NB. We produce the distance matrix
e3distances =: (0.8 spikeDistances) e3spikes

We use thresholding to generate our connectivity matrix:

e3connections =: e3distances < 2

And we turn the crank:

e3Lap =: calcLap e3connections
e3evs =: eigenvectors e3Lap
e3clustering =: 2 km (2 takeEigenvectors e3evs)

Finally, we can color code our rastergram by clustering:

pd 'reset'
pd 'type dot'
pd 'color blue'
pd 'title trial clustering'
pd spikesForPlotting (e3clustering = 0) # e3spikes

pd 'color red'
pd (#((e3clustering=0)#e3spikes)) spikesForPlotting ((e3clustering =   1) # e3spikes)

pd 'xcaption time'
pd 'ycaption trial'

pd 'show'

Example 3 Color Coded Rastergram



A Note On Simpler Clustering Algorithms

One can appreciate a family of clustering approaches by starting with Expectation Maximization: a two step iterative process which attempts to find the parameters of a probability distribution assumed to underly a given data set. When using this approach for clustering, one assumes that the data has hidden or missing dimensions: in particular, a missing label assigning each point to a cluster. Given some guess at those labels, you can calculate the parameters of the distribution. Using this estimate, you can modify the guess about the hidden labels. Repeat until convergence.

In practice, simplification over the general case can be found by assuming that the distribution is a "Gaussian Mixture Model," just a weighted combination of n-dimensional Gaussian distributions. Further simplifications manifest by applying ad-hoc measures of cluster "belonging" based on just the mean of each cluster (see fuzzy clustering) and finally just by assuming that objects belong to the cluster to which they are closest (as in k-means).

In all of these simplifications clustering refers to some sort of centroid of the clusters as a way of estimating which object belongs to which clusters. But some data sets have clusters which are concentric. For instance, if one cluster is points between 10 and 20 units from the origin and another is points between 50 and 60 units from the origin.

In this case, all such attempts will fail unless the space is transformed. Spectral clustering transforms the space by considering topological information: that is, what objects are near or connected to what other objects, and throwing away large scale information.

In the concentric example above, the thing that all points in each cluster have in common is they are near other points in their cluster. When our distributions have the property that the distance between some two points in cluster A may be greater than the distance between some point in A and some point in B, the spectral clustering gives a nice mechanism for identifying the clusters.



Throughout the above I refer to the occasional utility. Here are the definitions.

load 'stats/distribs'
load 'numeric'
load 'graphics/plot'

gausian =: dyad define
 'm s' =. x
 m + (s*(rnorm y))

uniform =: dyad define 
 'mn mx' =. x
 mn + ((mx-mn)*(runif y))

shuffle =: (#@:] ? #@:]) { ] 

hist =: dyad define 
  'min max steps' =. x
  'data' =. ((y<min) * (y>max)) # y
  'quantized' =. >. (steps * ((data - min) % (max-min)))
  'indexes' =. i. steps
  'stepSize' =. (max-min)%steps
  1 -~ ((indexes, quantized) #/. (indexes, quantized))

curtail =: }: 

histPlot =: dyad define 
  h =. x hist y 
  'min max nsteps' =. x 
  halfStep =. (max-min)%(2*nsteps)
  'plotXs' =. halfStep + steps (min, max, nsteps)
  'xcaption position; ycaption count; title example 1 data' plot (curtail plotXs) ; h 

unboxFrom =: >@:([ { ])

dot =: +/ . *

dir =: monad define 
  list (4 !: 1) 0 1 2 3

xyForPlotting =: (0&{ ; 1&{)@|:

ten =: monad define 



abs =: (+ ` - @. (< & 0))"0

nonzeros =: ((>&0@:abs@:]) # ])

asArray =:  ". ;. _2

concatSpikes =: dyad define
  maxSpikes =. (1 { ($ x)) >. (1{($y))
  pad =. (maxSpikes&{.)"1
  (pad x), (pad y)

spikeIndexes =: |:@:(( 1&{@:$@:], #@:[) $ (i.@:#@:]) )
spikesForPlotting =: verb define 
  0 spikesForPlotting y
  spikes =. ,y
  ii =:  ,spikeIndexes y
  nzii =. -. (spikes = 0)
  (nzii # spikes) ; (x+(nzii # ii))

rastergram =: monad define
  pd 'reset'
  pd 'type dot'
  pd 'color red'
  pd 'title rastergram'
  pd 'xcaption time'
  pd 'ycaption trial number'
  pd spikesForPlotting y
  pd 'show'

offsetY =: ] + ((#@:] , 1&{@:$@:]) $ (0: , [))

eye =: (],]) $ ((1: + ]) take 1:)

asOperator =: adverb define 
  m dot y



An implementation of k-means in J is comically terse, even in an exploded, commented form:

NB. This file contains an implementation of k-means, a clustering
NB. algorithm which takes a set of vectors and a number of clusters
NB. and assigns each vector to a cluster.
NB. Example Data
NB. v1 =: ?  10 2 $ 10 
NB. v2 =: ?  3 2 $ 10
NB. v3 =: ? 100 2 $ 50
NB. trivialV =: 2 2 $ 10
NB. Examples
NB. 3 km v3

NB. produce a permutation of y.
permutation =: (#@[ ? #@[)

NB. shuffle list
NB. shuffle the list
shuffle =: {~ permutation

NB. nClusters drawIds vectors
NB. given a list of vectors and a number of clusters
NB. assign each vector to a cluster randomly
NB. with the constraint that all clusters are
NB. roughly equal in size.
drawIds =: shuffle@(#@] $ i.@[)

NB. distance between two vectors as lists
vd =: +/@:(*:@-)

NB. Give the distance between all vectors in x and y
distances =: (vd"1 1)/

NB. return the list of indices of the minimum values of the rows.
minI =: i. <./

NB. Given x a list of centers and y a list of vectors,
NB. return the labeling of the vectors to each center.
reId =: ((minI"1)@distances)~

NB. canonicalize labels 
NB. given a list of labels, re-label the labels so
NB. that 0 appears first, 1 second, etc
canonicalize =: ] { /:@~.

NB. (minI"1 v1 distances v2 ) 

vsum =: +/"1@|:
vm =: (# %~ (+/"1@|:))

calcCenters =: (/:@~.@[) { (vm/.)

NB. ids kmOnce vectors This performs the heavy lifting for K-means.
NB. Given a list of integral ids, one for each vector in the N x D
NB. vectors, this verb calculates the averages of the groups implied
NB. by those labels and re-assigns each vector to one of those
NB. averages, whichever it is closest to.

kmOnce =: dyad define
  ids =. x 
  vectors =. y
  centers =. ids calcCenters vectors
  centers reId vectors

NB. Use the power adverb to iterate kmOnce until the labels converge
kmConverge =: ((kmOnce~)^:_)~

NB. nClusters km dataset k means.  Given a cluster count on the left
NB. and a vector data set on the right, return an array of IDs
NB. assigning each vector to a cluster.
NB. One can use `calcCenters` to recover the cluster centers.

km =: canonicalize@((drawIds) kmConverge ])

NB. Example Data
NB. v1 =: ?  10 2 $ 10 
NB. v2 =: ?  3 2 $ 10
NB. v3 =: ? 100 2 $ 50
NB. trivialV =: 2 2 $ 10

NB. Examples
NB. 3 km v3