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# Internal Utility Modules

During development of Owl, we find some utility modules are immensely handy. In this chapter, we share some of them. These are not the main feature of Owl, and perhaps you can implement your own version very quickly. But we hope to present how these features are used in Owl.

## Dataset Module

The dataset modules provides easy access to various datasets to be used in Owl, mainly the MNSIT and CIFAR10 datasets. You can get all these data in Owl by executing: Dataset.download_all (). The data are downloaded in the home directory, for example, ~/.owl/dataset on Linux.

### MNIST

The MNIST database of handwritten digits has a training set of 60,000 examples, and a test set of 10,000 examples. Each example is of size 28 x 28. It is a good starting point for deep neural network related tasks.

You can get MNIST data via these Owl functions:

• Dataset.load_mnist_train_data (): returns a triplet x, y, y'.

• x is a [60000, 784] ndarray (Owl_dense_ndarray.S.mat) where each row represents a [28, 28] image.
• y is a [60000, 1] label ndarray. Each row is an integer ranging from 0 to 9, indicating the digit on each image.
• y' is a [60000, 10] label ndarray. Each one-hot row vector corresponds to one label.
• Dataset.load_mnist_test_data (): returns a triplet. Similar to load_mnist_train_data, only that it returns the test set, so the example size is 10,000 instead of 60,000.

• Dataset.load_mnist_train_data_arr (): similar to load_mnist_train_data, but returns x as [60000,28,28,1] ndarray

• Dataset.load_mnist_test_data_arr (): similar to load_mnist_train_data_arr, but it returns the test set, so the example size is 10, 000 instead of 60, 000.

• Dataset.draw_samples x y n draws n random examples from images ndarray x and label ndarray y.

Here is what the dataset looks like when loaded into Owl:

# let x, _, y = Dataset.load_mnist_train_data_arr ();;
val x : Owl_dense_ndarray.S.arr =

C0
R[0,0,0]   0
R[0,0,1]   0
R[0,0,2]   0
R[0,0,3]   0
R[0,0,4]   0
...
R[59999,27,23]   0
R[59999,27,24]   0
R[59999,27,25]   0
R[59999,27,26]   0
R[59999,27,27]   0

val y : Owl_dense_matrix.S.mat =

C0  C1  C2  C3  C4  C5  C6  C7  C8  C9
R0   0   0   0   0   0   1   0   0   0   0
R1   1   0   0   0   0   0   0   0   0   0
R2   0   0   0   0   1   0   0   0   0   0
R3   0   1   0   0   0   0   0   0   0   0
R4   0   0   0   0   0   0   0   0   0   1
... ... ... ... ... ... ... ... ... ...
R59995   0   0   0   0   0   0   0   0   1   0
R59996   0   0   0   1   0   0   0   0   0   0
R59997   0   0   0   0   0   1   0   0   0   0
R59998   0   0   0   0   0   0   1   0   0   0
R59999   0   0   0   0   0   0   0   0   1   0

You can find the MNIST dataset used in training and testing a DNN in Owl:

let train network =
let x, _, y = Dataset.load_mnist_train_data_arr () in
Graph.train network x y |> ignore;
network

let test network =
let imgs, _, labels = Dataset.load_mnist_test_data () in
let m = Dense.Matrix.S.row_num imgs in
let imgs = Dense.Ndarray.S.reshape imgs [|m;28;28;1|] in

let mat2num x = Dense.Matrix.S.of_array (
x |> Dense.Matrix.Generic.max_rows
|> Array.map (fun (_,_,num) -> float_of_int num)
) 1 m
in

let pred = mat2num (Graph.model network imgs) in
let fact = mat2num labels in
let accu = Dense.Matrix.S.(elt_equal pred fact |> sum') in
Owl_log.info "Accuracy on test set: %f" (accu /. (float_of_int m))

### CIFAR-10

The CIFAR-10 dataset include small scale color images for more realistic complex image classification tasks. It includes 10 classes of images: aeroplane, automobile, bird, cat, deer, dog, frog, horse, ship, truck. It consists of 60,000 32 x 32 colour images in 10 classes, with 6,000 images per class. There are 50,000 training images and 10,000 test images.

Due to the limit of file size on Github, the training set is cut into 5 smaller batch. You can get CIFAR-10 data using Owl:

• Dataset.load_cifar_train_data batch: returns a triplet x, y, y'.

• The input paramter batch can range from 1 to 5, indicating which training set batch to choose.
• x is an [10000, 32, 32, 3] ndarray (Owl_dense_ndarray.S.arr). The last dimension indicates color channels (first Red, then Green, finally Blue).
• y is an [10000, 1] label ndarray, each number representing an image class.
• y' is the corresponding [10000, 10] one-hot label ndarray.
• Dataset.load_cifar_test_data (): similar to load_cifar_train_data, only that it loads test data.

• Dataset.draw_samples_cifar x y n draws n random examples from images ndarray x and label ndarray y.

Note that all elements in the loaded matrices and ndarrays are of float32 format.

The CIFAR10 dataset is similar to MNIST in training DNN:

let train network =
let x, _, y = Dataset.load_cifar_train_data 1 in
Graph.train network x y

## Graph Module

The Graph module in Owl provides a general data structure to manipulate graphs. It is defined as:

type 'a node =
{ mutable id : int
; (* unique identifier *)
mutable name : string
; (* name of the node *)
mutable prev : 'a node array
; (* parents of the node *)
mutable next : 'a node array
; (* children of the node *)
mutable attr : 'a (* indicate the validity *)
}

The attribution here is generic so that you can define your own graph where each node contains an integer, a string, or any data type you define. This make the graph module extremely flexible.

Graph module provides a rich set of APIs. First, you can build a Graph using these methods:

• node ~id ~name ~prev ~next attr creates a node with given id and name string. The created node is also connected to parents in prev and children in next. The attr will be saved in attr field.

• connect parents children connects a set of parents to a set of children. The created links are the Cartesian product of parents and children. In other words, they are bidirectional links between parents and children. Note that this function does not eliminate any duplicates in the array.

• connect_descendants parents children connects parents to their children. This function creates unidirectional links from parents to children. In other words, this function save children to parent.next field.

• connect_ancestors parents children connects children to their parents. This function creates unidirectional links from children to parents. In other words, this function save parents to child.prev field.

• remove_node x removes node x from the graph by disconnecting itself from all its parent nodes and child nodes.

• remove_edge src dst removes a link src -> dst from the graph. Namely, the corresponding entry of dst in src.next and src in dst.prev will be removed. Note that it does not remove [dst -> src] if there exists one.

• replace_child x y replaces x with y in x parents. Namely, x parents now make link to y rather than x in next field. Note that the function does note make link from y to x children. Namely, the next field of y remains intact.

Then, to obtain and update properties of a graph:

val id : 'a node -> int
(** id x returns the id of node x. *)

val name : 'a node -> string
(** name x returns the name string of node x. *)

val set_name : 'a node -> string -> unit
(** set_name x s sets the name string of node x to s. *)

val parents : 'a node -> 'a node array
(** parents x returns the parents of node x. *)

val set_parents : 'a node -> 'a node array -> unit
(** set_parents x parents set x parents to parents. *)

val children : 'a node -> 'a node array
(** children x returns the children of node x. *)

val set_children : 'a node -> 'a node array -> unit
(** set_children x children sets x children to children. *)

val attr : 'a node -> 'a
(** attr x returns the attr field of node x. *)

val set_attr : 'a node -> 'a -> unit
(** set_attr x sets the attr field of node x. *)

Similarly, you can get other properties of a graph use the other functions:

• indegree x returns the in-degree of node
• outdegree x returns the out-degree of node
• degree x returns the total number of links of x
• num_ancestor x returns the number of ancestors of x
• num_descendant x returns the number of descendants of x
• length x returns the total number of ancestors and descendants of x

Finally, we provide functions for traversing the graph in either Breadth-First order or Depth-First order. You can also choose to iterate the descendants or ancestors of a given node.

val iter_ancestors
:  ?order:order
-> ?traversal:traversal
-> ('a node -> unit)
-> 'a node array
-> unit
(** Iterate the ancestors of a given node. *)

val iter_descendants
:  ?order:order
-> ?traversal:traversal
-> ('a node -> unit)
-> 'a node array
-> unit
(** Iterate the descendants of a given node. *)

val iter_in_edges : ?order:order -> ('a node -> 'a node -> unit) -> 'a node array -> unit
(** Iterate all the in-edges of a given node. *)

val iter_out_edges : ?order:order -> ('a node -> 'a node -> unit) -> 'a node array -> unit
(** Iterate all the out-edges of a given node. *)

val topo_sort : 'a node array -> 'a node array
(** Topological sort of a given graph using a DFS order. Assumes that the graph is acyclic.*)

You can also use functions: filter_ancestors, filter_descendants, fold_ancestors, fold_descendants, fold_in_edges, and fold_out_edges to perform fold or filter operations when iterating the graph.

Within Owl, the Graph module is heavily use to facilitate the Computation Graph module.

TODO: Explain how it is used in CGraph.

TODO: Use examples and text, not just code.

## Stack and Heap Modules

Both Stack and Heap are two common abstract data types for collection of elements. They are also used in Owl code. Similar to graph, they use generic types so that any data type can be plugged in. Here is the definition of a stack:

type 'a t =
{ mutable used : int
; mutable size : int
; mutable data : 'a array
}

The stack and heap modules support four standards operations:

• push: push element into a stack/heap
• pop: pops the top element in stack/heap. It returns None if the container is empty
• peek: returns the value of top element in stack/heap but it does not remove the element from the stack. None is returned if the container is empty.
• is_empty: returns true if the stack/heap is empty

The stack data structure is used in many places in Owl: the filteri operation in ndarray module, the topological sort in graph, the reading file IO operation for keeping the content from all the lines, the data frame… The heap structure is used in key functions such as search_top_elements in the Ndarray module, which searches the indices of top values in input ndarray according to the comparison function.

## Count-Min Sketch

Count-Min Sketch is a probabilistic data structure for computing approximate counts. It is particularly ideal for use when the space is limited and exact results are not required. Imagine that we want to count how frequent certain elements are in a realtime stream, what would you do? An intuitive answer is that you can create a hash table, with the element as key and its count as value. The problem with this solution is that the stream could have millions and billions of elements. Even if you somehow manage to cut the long tail (such as the unique elements), the storage requirement is still terribly large.

Now that you think about it, you don’t really care about the precise count of an element from the stream. What you really need is an estimation that is not very far away from the true. That leaves space for optimising the solution. First, apply a hashing function and use h(e) as the key, instead of the element e itself. Besides, the total number of key-value pairs cab be limited. Towards the end, this approach can be summarised as three steps:

1. initialised an array of $$n$$ elements, each set to 0;
2. when processing one element $$e$$, increase the count of the hashed index: count[h(e)] += 1;
3. when querying the count for certain element, just return count[h(e)].

Obviously, this approach tends to give an overestimated answer because of the inevitable collision in hash table. Here the Count-Min Sketch method comes to help. It’s basic idea is simple: follow the process stated above, but the only difference is that now instead of maintaining a vector of length $$n$$, we now need to maintain a matrix of shape $$dxn$$, i.e. $$d$$ rows and $$n$$ columns. Each row is assigned with a different hash function, and when processing one element $$e$$, apply $$h_0, h_1, \ldots, h_d$$ to it, and make $$count[i][h_i(e)] += 1$$, for each $$i = 0, 1, 2, \ldots, d$$. At any time if you want to know the count of an element $$e$$, you again apply the same set of hash functions, retrieve the $$d$$ counts of this element from all the rows, and choose the smallest count to return. This process is shown in fig. 1 (Src).

In this way, the effect of collision is reduced in the counting. The reason is simple: if these different hashing functions are independent, then the probability that the same element leads to collision in multiple lines can be exponentially reduced with more hash function used.

Even though this method looks like just a heuristic, it actually provides an theoretical guarantee of its counting error. Specifically, we have two error bounds parameter: failure probability $$\sigma$$, and the approximation ratio $$\epsilon$$, and let $$s$$ be the sum of all counts stored in the data structure. It can be proved that with a probability of $$1-\sigma$$, the error between the the estimated count and the true count is $$\epsilon~s$$ at most. The detailed proof can be see in the original paper (Cormode and Muthukrishnan 2005).

Owl has provided this probabilistic data structure. It is implemented by Pratap Singh. Owl provides these interfaces for use:

module type Sig = sig
type 'a sketch
(** The type of Count-Min sketches *)

(** {6 Core functions} *)

val init : epsilon:float -> delta:float -> 'a sketch
(**
init epsilon delta initializes a sketch with approximation ratio
(1 + epsilon) and failure probability delta.
*)

val incr : 'a sketch -> 'a -> unit
(** incr s x increments the frequency count of x in sketch s in-place. *)

val count : 'a sketch -> 'a -> int
(** count s x returns the estimated frequency of element x in s. *)

val init_from : 'a sketch -> 'a sketch
(**
init_from s initializes a new empty sketch with the same parameters as s, which
can later be merged with s.
*)

val merge : 'a sketch -> 'a sketch -> 'a sketch
(**
merge s1 s2 returns a new sketch whose counts are the sum of those in s1 and s2.
Raises INVALID_ARGUMENT if the parameters of s1 and s2 do not match.
*)
end

TODO: a simple example of using it. (Perhaps also the benchmarking from blog)

CODE

Count-Min Sketch is a useful data structure when we are interested in the approximate counting of important objects in a group of things. One such application is to find heavy hitters. For example, finding out the most popular web pages given a large website access log with a long history. (DETAIL)

## Summary

Cormode, Graham, and Shan Muthukrishnan. 2005. “An Improved Data Stream Summary: The Count-Min Sketch and Its Applications.” Journal of Algorithms 55 (1): 58–75.