Chapter 3 Variable Types and Data Structures

In exploring the components that machine learning systems are built on, we stressed that different types of hardware are optimised to work with data and models stored in specific formats. Both are complex entities comprising a variety of elements that are organised into data structures such as data frames (representing tabular data) or standardised model formats like ONNX (ONNX 2021). The individual elements inside these data structures are variables of specific types such as integer numbers, floating point numbers and strings.

In this chapter we revisit the variable types we mentioned in Chapter 2, as well as string representations, and we discuss possible reasons to choose one over another for different classes of data (Section 3.1). We then give some notable examples of how variables are organised in data structures such as vectors and lists (Section 3.2.1), data frames (Section 3.2.2) and matrices (Section 3.2.3). Different choices of variable types (Section 3.3) and data structures (Section 3.4) represent different trade-offs both in terms of hardware support, as we saw in Chapter 2, and in terms of the computational complexity of the machine learning algorithms that will operate on them, as we will discuss in Chapter 4.

3.1 Variable Types

Machine learning software primarily deals with numbers that are the mathematical representation of the data we are modelling. Images can be represented using the values of the colour channels of each of their pixels; text can be encoded into strings, which are then converted to frequencies for particular words; sensor data are recorded as a time series. We can store each of them with different types of variables, each with pros and cons that we will discuss in Section 3.3.

3.1.1 Integers

Integer variables can be used to represent natural (\(\mathbb{N}\)) or integer numbers (\(\mathbb{Z}\)). They are often used to represent Boolean variables as indicators (also known as dummy variables) as follows.

More generally, they can be used to give a numeric representation to finite sets by mapping each element to a different integer number.

Enumerations in C or factors in R are constructed exactly in this way to minimise memory usage. However, this numeric representation is not suitable for modelling discrete variables because it makes parameter estimates dependent on the specific mapping between elements and integer numbers.7 Instead, we map discrete variables to their one-hot encoding:8 each element in the set is assigned an indicator variable that takes a value of 1 if the element is observed, and 0 otherwise.

The number of elements we want to represent determines how many bits of memory each integer will use: \(n\) bits allow for \(2^n\) distinct values. A single bit is enough for a Boolean value, although, in practice, it is usually stored using at least 1 byte. A finite set with \(k\) elements can be represented with \(\log_2 k\) bits: one-hot encoding side-steps this limitation by using indicator variables at the cost of using more memory.

Natural and integer numbers cannot be completely represented by integer variables: that would require an infinite number of bits. For this reason, programming languages provide integer variables of different sizes such as 8, 16, 32 and 64 bits. These sizes are all multiples of 8 bits because processors are optimised to work on bytes. The size of an integer variable determines the largest number it can represent. For instance, the largest (unsigned) natural number we can represent in 32 bits is \(2^{32} - 1 \approx 4.29 \times 10^9\), and the largest (signed) integer number is \(\pm 2^{31} - 1 \approx \pm 2.14 \times 10^{9}\) because 1 bit is reserved for encoding the sign. (If that bit is set to zero, the number is considered to be positive.) Numbers that are outside of this range are said to overflow, meaning that their bit representation is larger than the size of the integer variable and thus overflows the memory reserved for that variable.

For the sake of the example, consider the natural number 3134 represented as a 16-bit unsigned integer variable.

It’s easy to check that this representation is equivalent to the “natural” one, they just use different bases: \[\begin{equation*} 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^{10} + 2^{11} = 3134. \end{equation*}\] The largest number that can be represented in 16 bits is \(2^{16} - 1 = 65535\), and 3134 is comfortably smaller than that: we can easily see that the 4 most-significant bits that represent the 4 largest powers of 2 (\(2^{15}\), \(2^{14}\), \(2^{13}\) and \(2^{12}\)) are all equal to zero. Now consider a much larger number: 247586.

Again, we can easily check that \[\begin{equation*} 2^1 + 2^5 + 2^8 + 2^9 + 2^{10} + 2^{14} + 2^{15} + 2^{16} + 2^{17} = 247586. \end{equation*}\]

Unfortunately, 247586 is larger than 65535 and cannot be represented in 16 bits. If we try to store it in 16 bits, we overflow: the integer variables will contain only the 16 least-significant bits representing the powers from \(2^0\) to \(2^{15}\). The bits corresponding to \(2^{16}\) and \(2^{17}\) will be silently dropped, and the integer variable will contain the number \[\begin{equation*} 2^1 + 2^5 + 2^8 + 2^9 + 2^{10} + 2^{14} + 2^{15} = 50978. \end{equation*}\]

In the case of signed integers, the bits that overflow will overwrite the sign bit, and result in the integer variable storing a number that may be incorrect in both sign and absolute value. Consider again the number 247586, this time represented as a 16-bit signed integer variable.

The first 15 bits of the binary representation are stored correctly, the bit corresponding to \(2^{15}\) overwrites the sign bit, and the last two bits corresponding to \(2^{16}\) and \(2^{17}\) are again silently dropped. As a result, the integer variable will contain the number \[\begin{equation*} - (2^1 + 2^5 + 2^8 + 2^9 + 2^{10} + 2^{14}) = -18210. \end{equation*}\]

As an exception to the rule, R represents a missing integer value (NA) with the largest negative signed integer for a given precision. In Python, Pandas uses masked arrays for the same purpose and keeps a separate Boolean variable that indicates whether the integer is a missing value (denoted pandas.NA). NumPy does not support missing values for integer variables.

Range limitations aside, integer variables allow exact computer arithmetic: their bit representation coincides with the mathematical representation of integer and natural numbers in base 2, so there is no rounding or loss of precision.

3.1.2 Floating Point

Floating point variables are used to represent real (\(\mathbb{R}\)) and complex numbers (\(\mathbb{C}\)), the former with a single variable and the latter with two (one for the real part, one for the imaginary part). Each variable is composed of four parts: the sign \(S\), the bias or offset \(O\), an exponent \(E\) and a mantissa \(M\). The floating point representation of a real number \(x\) is then \[\begin{equation*} x = (-1)^S * (1 + M) 2^{E + O}. \end{equation*}\] The overall size of the variable in bits is typically one of 16, 32, and 64 bits, often called half-precision, single-precision, double-precision. The number of bits assigned to each of the exponent (after adding the offset) and the mantissa depends on what encoding is used; the sign is always stored in a single bit. The variables defined in the IEEE 754 standard (Overton 2001) reserve:

precision exponent mantissa
half (16 bit) 5 bits 10 bits
single (32 bit) 8 bits 23 bits
double (64 bit) 11 bits 52 bits

The value of the offset is determined as \(O = 2^{|E| - 1} - 1\) where \(|E|\) is the size of the exponent in bits.

The alternative “brain” format devised by Google in the process of developing TPUs (see Section 2.1.1) typically has size 16 bits and is known as “bfloat16”. It works in the same way as the IEEE 754 floating point, so we will not discuss it further; the only difference is that it allocates 8 bits to the exponent and 7 bits to the mantissa.

What does this mean in terms of binary representation? Consider the number 435.25. In the usual scientific notation, which is in base 10, we can write it as \(4.3525 \times 10^2\). If we do the same in base 2, the scientific notation becomes \(1.7001953125 \times 2^8\). The exponent is \(8\), and the mantissa is \(0.7001953125\). As a half-precision floating point variable, 435.25 then has the following binary representation:

The exponent is stored after adding the offset: \[\begin{equation*} 8 + (2^{5 - 1} - 1) = 23 = 2^0 + 2^1 + 2^2 + 2^4. \end{equation*}\] The resulting number is treated as unsigned, regardless of whether the original exponent was positive or negative. If the sign is stored in the most significant bit of the variable, the exponent is adjusted with the offset, and the mantissa is stored in the least significant bits, we can compare floating point numbers just by ranking their binary representations, which can be done efficiently with hardware instructions.

The mantissa is \[\begin{equation*} 2^{-1} + 2^{-3} + 2^{-4} + 2^{-7} + 2^{-8} + 2^{-10} = 0.7001953, \end{equation*}\] which differs from \(0.7001953125\) by \(1.25 \times 10^{-8}\). This difference is known as the floating point error arising from the limits in the precision that can be achieved with the number of bits of the mantissa. The only numbers that can be represented exactly are those that factorise into the powers of 2 available in the exponent and in the mantissa. This obviously excludes all numbers with an infinite number of digits, such as \(\pi\), \(e\) or \(1/3\).

What is the range of floating point numbers? The largest number (positive or negative) that we can represent is limited by the size of the exponent: with \(|E|\) bits we can represent up to \(2^{|E|}\) exponents. The offset ensures that the available exponents are equally divided between positive and negative numbers ranging from \(-2^{|E| - 1} - 2\) to \(2^{|E| - 1} - 1\) due to the offset.

precision smallest exponent largest exponent
half \(-(2^{4} - 2) = -14\) \(2^{4} - 1 = 15\)
single \(-(2^{7} - 2) = -126\) \(2^{7} - 1 = 127\)
double \(-(2^{10} - 2) = -1022\) \(2^{10} - 1 = 1023\)

This range is smaller than the theoretical \(2^{|E|}\) values it could potentially contain because some combinations of bits are reserved for special classes of numbers:

  • Zero is encoded with the exponent field and the mantissa filled with 0s.
  • Positive and negative infinity (+Inf, -Inf) are encoded with the exponent field filled with 1s and a mantissa filled with 0s.
  • Irrepresentable numbers (usually denoted NaN) are encoded with the exponent field filled with 1s and at least one non-zero bit in the mantissa. Different patterns of bits in the mantissa are used for different types of irrepresentable numbers: the most common is the missing value identifier NA. Typically, NaN arises from dividing a number by zero or by trying to apply a mathematical function to a value outside its domain, for instance, taking the logarithm of a negative number.
  • Subnormal numbers, that is, numbers that are too small to be written in binary scientific notation with the available exponents. In other words, their leading exponent is smaller than the smallest available exponent. They are encoded with the exponent field filled with 0s. These numbers have reduced precision because they effectively use only part of the mantissa. As an example, \(2^{-10} \times 2^{-14} \approx 5.96 \times 10^{-8}\) is represented as follows in half precision:

How coarse is floating point rounding? For any given precision, that depends on the magnitude of the number. As we saw in the example above, the mantissa can encode only so many significant decimal digits: \(\log_{10}(2^{10}) \approx 3\) digits for half-precision, \(\log_{10}(2^{23}) \approx 7\) for single precision, \(\log_{10}(2^{51}) \approx 16\) for double precision. This effectively creates a grid of values that can be represented exactly, and any other number is rounded to the nearest number that can be represented exactly or to +Inf/-Inf. The grid becomes coarser, in absolute terms, as the exponent becomes larger. Consider a number like 0.0002 that is small for a half-precision variable:

The exponent is \[\begin{equation*} -13 + (2^{5 - 1} - 1) = 2^1 \end{equation*}\] and the mantissa is \[\begin{equation*} 2^{-1} + 2^{-3} + 2^{-7} + 2^{-8} + 2^{-9} = 0.638671875, \end{equation*}\] which gives \[\begin{equation*} 1.638671875 \times 2^{-13} \approx 0.000200033. \end{equation*}\] Increasing this number by the least possible amount by adding \(2^{-10}\) and decreasing it by the same amount shows that the nearest numbers that can be represented in half precision are \(\approx 0.000200033 \pm 1.19 \times 10^{-7}\).

Now consider a relatively large number (for half precision) like 10002:

The exponent is \[\begin{equation*} 13 + (2^{5 - 1} - 1) = 28 = 2^2 + 2^3 + 2^4 \end{equation*}\] and the mantissa is \[\begin{equation*} 2^{-3} + 2^{-4} + 2^{-5} + 2^{-9} = 0.220703125, \end{equation*}\] which gives \(1.220703125 \times 2^{13} = 10000\). The closest numbers that can be represented in half precision are 9992 and 10008: all the numbers in between are rounded. This leaves an interval of \(\pm 8\) around 10000. For large enough numbers, floating point variables cannot even represent integer numbers without rounding!

How can we keep the errors introduced by floating point rounding in check? Errors compound across operations, and machine learning models typically perform large numbers of operations compared to the size of their inputs. (More on that in Chapter 4.) Fortunately, probability theory and statistics have historically standardised computations to work on the logarithmic scale to make closed-form mathematical derivations easier. Working with the logarithmic transforms of floating point numbers reduces the chances that large numbers will overflow to +Inf/-Inf or that small numbers will be rounded down to zero. In the case of numbers with subnormal floating point representations, we also retain better precision because their logarithm will not be subnormal. This is particularly important in the common case of summing up large numbers of log-probabilities. Working with numbers on the same scale (that is, they have similar exponents) also helps in avoiding catastrophic losses in precision. When operations involve numbers on very different scales, the difference in the granularity of the floating point rounding may cause the result to have unacceptably large errors even though all the operands are accurate. As an extreme example, consider adding 10002 and 0.0002: the result would be 10000, the closest floating point number in half precision! A similar issue is catastrophic cancellation, which may happen when subtracting two floating point numbers that are very close to each other.

Unlike integer arithmetic, floating point arithmetic is not exact because of the impact of floating point rounding. The results of operations involving floating point variables can differ in many ways from the mathematical operations they implement, even in common scenarios. It is easy to demonstrate with a simple recurrence such as \[\begin{align*} x_0 = 4, x_1 = 4.25, x_{n + 1} = 108 - \left(815 - \frac{1500}{x_{n - 1}}\right) \frac{1}{x_n}, \end{align*}\] which converges to 100 in double precision even though the true limit in \(\mathbb{R}\) is 5 (Rump 2020b). This can happen even if all the operands are exactly representable, as proved in (Rump 2020a). Some effects of this discrepancy are:

  • Numbers that should be equal are not equal. We should always compare numbers with a tolerance that is a function of the floating point precision we are using. The default in R is the square root of the smallest representable number, obtainable as sqrt(.Machine$double.eps).
sqrt(2) * sqrt(2) == 2
## [1] FALSE
all.equal(sqrt(2) * sqrt(2), 2, tol = sqrt(.Machine$double.eps))
## [1] TRUE
  • Conversely, numbers that should not be equal may end up being equal.
1e99 == 1e99 + 1
## [1] TRUE
1 - 1e-20 == 1
## [1] TRUE
  • The order in which operations are performed matters, even when the mathematical operations or functions they implement are commutative and/or associative. Structuring code so that key computations are implemented only once and therefore ensuring that operations are always performed in the same sequences is the best way to prevent this issue.
print(0.6 + 0.7 + 0.8, digits = 20)
## [1] 2.0999999999999996447
print(0.8 + 0.7 + 0.6, digits = 20)
## [1] 2.1000000000000000888
  • The order in which operations are performed matters also because intermediate results may underflow to zero or overflow to +Inf/-Inf unless we reorder operations to prevent that from happening.

  • Working on a log-scale is the best option when dealing with the small probabilities that often arise from multivariate distributions or from a large number of data points. Otherwise, the final result is likely to underflow to zero.

probs = runif(10^2, min = 10^-6, max = 10^-3)
## [1] 0
exp(0.5 * sum(log(probs)))
## [1] 1.33e-169

3.1.3 Strings

Strings are sequences of characters encoded in binary form and stored into variables. Their binary format varies, but it typically is UTF-8 on Linux and MacOS X and UTF-16 on Windows. Both are Unicode standards (Unicode 2021) that use between 1 and 4 bytes to encode each character and support many alphabets, mathematical symbols and pictograms such as emoji.

In the context of machine learning software, character strings are typically only encountered as input data in natural language processing (NLP) applications. In other settings, they are used as human-readable labels for the items in a set and can be represented using integers as we saw in Section 3.1.1. In fact, they are eventually given a numerical representation even in NLP in order to feed them to algorithms such as word2vec (Rong 2014), GLOVE (Pennington, Socher, and Manning 2014) and BERT (Devlin et al. 2019). In NLP, strings are also preprocessed taking into account their meaning and their grammatical and syntactical properties to facilitate later analyses. For instance:

  • Common words that do not add meaning to a sentence, often called stopwords, are removed to reduce the dimensionality of the data.
  • Words may be stemmed, that is, different words may be reduced to their common stem after removing suffixes and prefixes to identify which are in fact the same word.
  • Words may be tagged with their syntactic role.
  • Words may be normalised by making all characters lower-case and sometimes by removing accents and diacritics as well. Complex, composite characters can be encoded in different ways in both UTF-8 and UTF-16, and transforming them into their canonical form is essential to identify unique words correctly.
  • Extraneous characters such as punctuation, hyphenation and numbers may be removed as non-informative. Abbreviations and acronyms may be expanded to make explicit the words they correspond to. Similarly, emoji may be replaced by a textual description.

A detailed treatment of these topics is beyond the scope of this book, and we refer the reader to monographs such as (Aggarwal 2018) and to the documentation of relevant software libraries such as Spacy (Explosion 2021) and NLTK (NLTK Team 2021).

3.2 Data Structures

Data structures are particular ways of organising variables of one or more types for effective and efficient processing. Different data structures will be most effective for different operations or different algorithms. We will discuss both aspects further in Section 3.4 and later in Chapter 4, characterising memory and computational efficiency in terms of space and time complexity. Here we will only cover those data structures that are foundational for machine learning software, referring the reader to other excellent resources (Brass 2008; Cormen 2013) for a broader coverage of the topic.

Why use data structures? Firstly, they make code more compact by allowing us to abstract away basic variable manipulations that would otherwise be repeatedly implemented in different places. Our code will be clearer and most likely have fewer bugs as a result. Secondly, data structures tell the software how particular groups of variables belong together, both in terms of how they are laid out in memory and how we operate on them. This makes it possible for the software we write to be compiled or interpreted (see Section 6.1) to operate efficiently on the variables contained in the data structures. Thirdly, the information that particular groups of variables belong together will be useful to developers working on our code. Those variables may describe the parts of a single mathematical object or real-world entity, they may have the same semantic meaning or they may have attached metadata that can be used for interpretation and debugging purposes: all facts that are useful to know when reading and developing code.

3.2.1 Vectors and Lists

The most fundamental data structures in machine learning software are vectors and lists. Both can contain any type of variable, and are defined by their length (the number of elements they contain). Their conceptual structure is shown in Figure 3.1.

A schematic view of the logical structure and the memory layout of vectors (left) and lists (right).

Figure 3.1: A schematic view of the logical structure and the memory layout of vectors (left) and lists (right).

Vectors are homogeneous data structures holding sequences of variables of the same type. The variables are stored sequentially in a single block of memory, so vectors can be accessed with a single memory access using a pointer to their first element. (A pointer is itself a variable that contains a memory address.) Reading the variables stored in a vector is trivial because all elements occupy the same number of bytes in memory: the \(i\)th element is located at a memory address that is that of the first element plus \(i\) times the variable type size. Copying the whole vector is also trivial, since it is stored as a single block of memory. The same is true for subsets of variables that are adjacent to each other within a vector.

Lists, on the other hand, are heterogeneous data structures that can contain different kinds of elements. They essentially act as vectors of pointers to arbitrary data structures or variable types. Therefore, each element in a list can be anything: a single variable of some type, a vector of any length, a second list, a matrix, etc. However, this means that accessing the elements of a list is less trivial since we need to locate each element and access it separately. However, copying the list and subsetting it can be easier: if we do not need to modify the contents of its elements, we can just copy (all or a subset of) the pointers to the elements to create a new list. This is called a shallow copy, and can significantly reduce memory use. However, we must duplicate the elements as well if we need to modify them later in the new list in order to avoid altering the original list they are attached to. Copying both the list and its elements is called a deep copy. In contrast, subsetting vectors requires a deep copy in the general case. Shallow copies are only possible when copying a whole vector or when subsetting a slice of adjacent elements.

Storing variables into vectors makes vectorised computations possible: a function can be applied independently to each element of the vector, potentially leveraging hardware’s SIMD and FMA instructions to achieve instruction- and data-level parallelism as we discussed in Section 2.2. If the return value of the function is a scalar, the results can be saved in a second vector of the same length. Otherwise, the results can be saved in a dense matrix (Section 3.2.3) or in a data frame (Section 3.2.2) in which each row or column contains the return values for a single input element. Vectorised computations are also possible for lists using thread-level parallelism, assuming that the function can handle all the types of variables stored in the list. Its outputs would then be stored in a second list regardless of whether each of them is a scalar or not.

3.2.2 Representing Data with Data Frames

A data frame is a heterogeneous two-dimensional data structure with columns of potentially different types. Its primary task is storing tabular data and the associated metadata, such as column- and row-names. The implementations in Julia (DataFrame.jl) and Python (Pandas and Scikit-Learn (Scikit-learn Developers 2022)) have been heavily inspired by R data frames: they only have minor differences in their semantics. The most notable is that operations on two data frames will match cells by position in R and Julia (regardless of row- and column-names) and by row- and column-names in Python (regardless of the cell positions).

A schematic view of tabular data (bottom right) encoded as a data frame (left, top).

Figure 3.2: A schematic view of tabular data (bottom right) encoded as a data frame (left, top).

The fundamental structure of a data frame is that of a list: each column in the tabular data is a vector that is stored as an element along with its own metadata as shown in Figure 3.2. Therefore, each column is stored in a separate block of memory, and there are no constraints on the types of variables that can be stored in different columns. In addition, a data frame typically contains its dimensions and the labels of the rows and of the columns as metadata, making it possible to access its contents as we would with a table. The dimensions are the number of rows and columns of the data frame. The labels of the columns (called “column names” in R) can be used to access them by their names instead of by their positions in the data frame, which improves the readability of code and thus our ability to debug it. It makes the code invariant to the data layout as well. The labels of the rows (“row names” in R) serve the same function but are not used as often: they usually have no practical use in the common case in which we assume that data points are independent and identically distributed.

Data frames make it efficient to operate on columns. Creating a new data frame with a subset of columns is like subsetting a list, with the additional step of carrying over row and column labels as needed. Copying it can be done efficiently with a shallow copy. Adding a column to a data frame is similar: we perform a shallow copy into a new data frame with an empty slot in which we can insert the vector storing the column’s values. Applying a function to each column of a data frame can be vectorised and performed in parallel, and the appropriate method can be called for each column in the case of generic functions.

However, operating on rows is not efficient in most cases. Adding or removing data points involves modifying the length of each column, which will likely involve copying the chosen data points to newly-allocated vectors of the appropriate length.

3.2.3 Dense and Sparse Matrices

A matrix is a homogeneous two-dimensional data structure holding a grid of variables of the same type arranged into rows and columns. It is the programming construct that represents the mathematical objects of the same name studied in linear algebra. Matrices can be dense or sparse. Most of the elements of dense matrices are informative (that is, non-zero cells) and therefore must be stored in the data structure. On the other hand, most of the elements of sparse matrices are equal to zero so we can save considerable amounts of memory by storing only the locations and the values of the few non-zero elements. We will cover the trade-off between speed and memory use for these two types of matrices in Section 4.5.2 while discussing computational complexity.

In Python, dense matrices are implemented in NumPy as a special case of multidimensional arrays along with vectors (Harris et al. 2020). The same is true in R. In both cases, the data structure encoding a multidimensional array comprises the pointer to the first element of the array; the variable type of the elements; and the dimensions of the array, which determine its shape (Figure 3.3). The dimensions and the variable type of the elements determine the strides: the number of bytes skipped in memory to proceed to the next element along a given dimension. They are pre-calculated and stored in NumPy but not in R. On the other hand, R arrays contain labels for their dimensions (row and column names in the case of matrices). The elements are stored as a vector, typically in column-major order: the columns of the matrix are concatenated starting from the left-most one.

A schematic view of a dense matrix (left) encoded as a multidimensional array with variables stored in column-major format (right).

Figure 3.3: A schematic view of a dense matrix (left) encoded as a multidimensional array with variables stored in column-major format (right).

Storing dense matrices as a list of columns, or as a list of rows, is typical in C and but it is less common in higher-level languages, where we can use data frames for the same purpose.

The fact that multidimensional arrays store their dimensions as metadata allows three types of operations on their elements. The first is vectorised operations in which a function is applied individually to each element. The second is what is called broadcasting in Python and Julia and recycling in R: when a function operates on two arrays with different dimensions, the shorter array is repeated (that is, virtually concatenated to itself) to make the shapes of the operands match. The third is marginalisation or reduction: aggregating elements across one or more dimensions of an array, for instance, by summing or averaging them, to produce a second array with fewer dimensions.

Sparse matrices are supported in R through the Matrix package (Bates and Maechler 2021) and in Python through the SciPy package (Virtanen et al. 2020). For brevity, we will only illustrate in detail the compressed sparse column data structure that is the most widely used in both packages. Consider the sparse matrix shown in Figure 3.4 along with its compressed representation from the Matrix package. The three vectors in the data structure contain, from top to bottom: the start and end indexes of each column (\(C\)), the row of each non-zero cell in the matrix (\(R\)) and its value (\(V\)). This representation assumes that the non-zero cells are stored in position order, starting from the top-left cell, moving down within each column, and considering columns from left to right.

A schematic view of a sparse matrix (left) and its compressed sparse column (right) format representation.

Figure 3.4: A schematic view of a sparse matrix (left) and its compressed sparse column (right) format representation.

Say, for instance, that we would like to read the value of the cell (2, 3) in the matrix from the data structure. The required steps are:

  1. Use the column delimiters in \(C\) to find which subset of \(R\) and \(V\) to read. The \(i\)th column of the matrix starts at the index stored in the \(i\)th element of \(C\) (\(C[3] = 3\)) and ends by the index stored in the \((i + 1)\)th element of \(C\) (\(C[4] = 5\)), where the next column starts. This implies that there are \(5 - 3 = 2\) non-zero elements in the third column. If the start and end indexes are identical, there are no non-zero cells in the column.
  2. We read the two row coordinates stored in \(R[3]\) and \(R[4]\).
    1. If we do not find the row coordinate we are looking for, the cell has value zero.
    2. Otherwise, the value of the cell will be stored in the element of \(V\) that has the same index as the row coordinate. In our case, the row coordinates of the non-zero elements of the third column are \(R[3] = 2\) and \(R[4] = 3\). Row coordinate \(2\) is in \(R[3]\), so we can read the corresponding cell value from \(V[3]\).

Other data structures for sparse matrices include the compressed row column, which is identical to the above save that the roles of \(R\) and \(C\) are reversed and the coordinate list (called the “triplet format”), which stores row and column coordinates directly in \(R\) and \(C\).

3.3 Choosing the Right Variable Types for the Job

The floating point format used to represent real numbers in Section 3.1.2 has a more complex structure than the fixed point format for integer variables in Section 3.1.1. Intuitively, this might suggest that the same operation is more efficient on integer variables than on floating point variables. This is not usually the case for two reasons. Firstly, high-level languages like Python and R have additional checks to deal with integer overflow. In R, integers are stored using 32 bits and they are either replaced with NaN or transformed into double-precision floating point variables when they overflow. In base Python, integers are stored with arbitrary precision: their size is extended as needed to prevent them from overflowing. Pandas (McKinney 2017) integer variables have size 64 bits, and NumPy (Harris et al. 2020) provides integer variables in sizes 8, 16, 32 and 64 bits: both can overflow and, unlike in R, are not replaced with NaN. Consider the following vector inner product benchmark in R:

floats.vec1 = rnorm(2 * 10^7)
floats.vec2 = rnorm(2 * 10^7)
integers.vec1 = sample(10, 2 * 10^7, replace = TRUE)
integers.vec2 = sample(10, 2 * 10^7, replace = TRUE)

microbenchmark(integers.vec1 %*% integers.vec2,
               floats.vec1 %*% floats.vec2, times = 200)

On average, the inner product takes 196.7% longer with integer vectors than it does with double-precision floating point vectors on a 7th-generation Intel Core processor. The same benchmark in Python and NumPy is shown below, and the results are similar: the inner product takes 40.5% longer with integer vectors.

import timeit
import numpy as np


float_vector1 = np.random.normal(0, 1, 2 * pow(10, 7))
float_vector2 = np.random.normal(0, 1, 2 * pow(10, 7))

int_vector1 = np.random.choice(10, size=2 * pow(10, 7))
int_vector2 = np.random.choice(10, size=2 * pow(10, 7))

def product_int():, int_vector2)

def product_float():, float_vector2)

print("Inner product with int by", ITERATION, "iteration, avg:",

print("Inner product with float by", ITERATION, "iteration, avg:",

Secondly, it should be apparent from Section 2.1.1 that in recent years much effort has been put into improving hardware support for floating point numbers. CPUs, GPUs and TPUs have all been optimised to handle single- and double-precision floating point variables with SIMD and FMA instructions as much as they have been optimised to handle integer variables, if not more. Therefore, depending on the available hardware and on the ability of compilers to leverage it, using floating point variables may lead to faster code when using low-level languages. However, whether that will be the case for a specific machine learning software depends on the exact combination of hardware and software used and can only be ascertained by benchmarking it. Matching software and hardware was a key point in Section 2.2 and Section 2.4.

The size of the variables also matters: we saw in Section 2.1.2 how faster forms of memory are smaller, and how copying data between different types of memory can impact operational intensity. We should always choose the smallest size of integer or floating point variables that we can handle with SIMD and FMA hardware instructions, and that has a large enough range to represent the numbers we are working with. The effect on performance is noticeable even in the simple Python benchmark above: reducing the size of the floating point and integer variables in the vectors from 64 bits (the default) to 32 bits and then to 16 bits produces interesting patterns in the normalised running times.

variable type 64 bits 32 bits 16 bits
floating point 100% 54.1% 887.1%
integer 100% 62.2% 61.6%

Reducing the size of both floating point and integer variables from 64 to 32 bits improves the speed of the inner product by a factor of 1.5–2, as we would expect. Reducing the size of the variables further to 16 bits provides only marginal benefits for integer variables but, surprisingly, slows floating point variables by a factor of nearly 9. That is a strong indication that we are unable to leverage SIMD and FMA instructions as we do for integers of the same size!

Size matters even more in the case of floating point variables. As we noted in Section 3.1.2, the smaller the precision, the larger the floating point errors are likely to be. They also propagate with each operation and compound each other. This can become a critical issue with variables that are involved in most of the steps of an algorithm, such as the accumulator variables used to calculate a sum or product of a series of values, and with those that are rescaled to predefined ranges with other computed quantities. An example from classical statistics is computing the empirical correlation between two vectors:

  1. We compute the average of each vector, which we store in two accumulator variables.
  2. We compute the variance of each vector by summing up the squared differences from its average, which we store in two more accumulator variables.
  3. We do the same with the cross-product of the differences to compute the covariance, which is an additional accumulator variable.
  4. We divide the covariance by the square root of the product of the variances.

Each of these steps can potentially build up an error large enough to produce a correlation coefficient that is either greater than 1 or less than -1. Floating point errors compound and propagate from the means to the variances and the covariances, affecting the final division through the accumulator variables that store them. In such a situation, we should store accumulator variables with a higher precision than the variables they are accumulating to limit the magnitude of the errors of the individual operations: for instance, we should store the average of numbers stored in single precision as a double precision variable. Using FMA instructions may also help because, as we noted in Section 2.1.1, they operate at higher precision and only round their final result. Choosing the scale of the numbers being accumulated may also help by keeping all variables in a range that is not prone to overflow or underflow. Keeping all variables on the same scale also prevents catastrophic loss of precision, particularly in multiplications and divisions. This is the reason why so much numeric software works with quantities on log-scales: large numbers are reduced in magnitude and do not overflow or lose precision easily, and small numbers become large negative numbers instead of underflowing or losing precision.

Last but not least, floating point rounding may be unacceptable for legal reasons in some applications, particularly in finance and accounting. Fixed point integers may be used instead, with the convention that the smallest possible amount of currency (say, 1/100th of 0.01) is taken as the unit value. A rare open-source example of this approach in the commercial world is Oanda’s libfixed (Oanda 2018) library.

3.4 Choosing the Right Data Structures for the Job

The choice of data structures can have an even larger impact than that of variable types because it determines memory access patterns. Operational intensity depends crucially on efficient memory use and access, as we argued in Sections 2.1.2 and 2.2.

Lists can be more memory efficient than vectors if we need to repeatedly subset them, for example, because we need to work on combinations or permutations of their values. If shallow copies are acceptable, lists are faster to duplicate as well because we do not have to allocate memory for and copy their elements. If shallow copies are not acceptable, then vectors are faster to copy because all their elements are stored as a single block of memory and can be copied with a single operation. And they are not less memory efficient, since their size is determined by their length. In fact, lists use more memory than vectors because they contain pointers to each element in addition to the elements themselves: the difference can be significant if the elements are small overall. These considerations are important for optimising performance given the effects on memory latency discussed in Sections 2.1.2 and 2.2.

We can make similar considerations for data frames and matrices, since data frames essentially behave as lists whose elements are the columns storing the variables in the tabular data.

Ideally, we should choose which data structures to use in our code taking into account what data structures are used in the libraries and in the other software that are part of the machine learning pipeline. If different parts of the pipeline encode data and models in different ways, we will be forced to convert between them, which is inefficient and increases memory use. For instance, R typically imports data as data frames. However, the underlying BLAS and LAPACK code that powers many models (all linear regressions among them) requires data to be stored as dense matrices in column-major format. Converting a data frame into a matrix requires copying all the data into a single memory block, column by column, which doubles memory use and wastes processor time.

We should also choose data structures based on how the algorithms that use them access their contents. Data that are processed together should be stored together to allow algorithms to perform as few separate memory accesses as possible. For instance, if we mostly process whole columns in tabular data, then a data frame is ideal because a single column can be efficiently read from memory as a single memory block. However, a data frame makes memory access very inefficient if we need to process individual rows in various combinations because each variable in a row is stored in a separate memory block and because we need to access all variables to read each row. If the data are homogeneous, storing them in a dense matrix with cells stored in row-major order is a better choice.


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  1. If we use the colour in a regression model, the effect of “green” on the response will be twice that of “red”, which clearly does not make any sense since the numbers associated with the colours are arbitrary.↩︎

  2. One-hot encoding is a particular case of what are known as contrasts in statistics. Since they are collinear, we usually drop one before using them in a model.↩︎