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What are Tensors | Tensor In-depth Explanation | Tensor in Machine Learning — Transcript

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0:00Hello guys, welcome to my YouTube
0:01channel. This is 100 Days of Machine
0:03Learning and today is day 11. So, 10
0:07days have passed and we have discussed
0:10many things, mostly related to the "why
0:12" and "what" parts of machine learning.
0:16The "why" and the "what," right? But
0:19from now on, from today onwards, we are
0:21going to focus on the "how" parts. How
0:23does machine learning happen? We are
0:25starting practical machine learning
0:26from today. And our topic for today is
0:29a very important one, because it is by
0:31using this that machine learning is
0:33able to function. Okay? Today we are
0:36going to study about tensors. Okay? I
0:38hope you have heard the name at some
0:40point. What are tensors? And why are we
0:44studying tensors today? I will answer
0:46these two questions for you very
0:47quickly. First, let me answer why we
0:49are studying tensors. First of all.
0:51Right? We could have studied something
0:53else. But why is the first thing we
0:54started in practical machine learning
0:56tensors? Let's discuss that. So, a
0:59tensor is basically a data structure. I
1:03mean, if I explain it to you in the
1:05simplest terms, a tensor is nothing but
1:07a data structure. A data structure
1:09means a way to store data. Now, why are
1:11we studying tensors first? The reason
1:14is that today, all machine learning
1:17systems and all the leading libraries
1:19like Scikit-learn and TensorFlow use
1:21the tensor as their most basic data
1:24structure. Meaning, whatever machine
1:27learning problem you are solving, in
1:29the machine learning space or the deep
1:31learning space, you will have to deal
1:33with tensors. So, as a beginner, I
1:35honestly skipped this topic. And later
1:38I realized that things wouldn't work
1:40without studying tensors. So, so that
1:42you don't make this mistake, we are
1:44starting with tensors. What you learn
1:47today will not only help you in machine
1:49learning but will also help you in deep
1:51learning. Okay? So, it is a very
1:53general topic. A very important topic.
1:55Let's start with the topic. Okay? So,
1:57one more thing I forgot to mention.
2:00Tensors are so important that the
2:02number one deep learning library today,
2:05Google's TensorFlow, is named after
2:08tensors. So, trust me, it is a very
2:09important concept. Okay? So, yeah,
2:11anyways, what is a tensor? Let's start
2:13the discussion from there again. A
2:14tensor is nothing but a data structure.
2:17Data structure means a way to store
2:19data. Right? And if we break down
2:22tensors into even simpler terms, a
2:24tensor is just a container where you
2:25store numbers. That's it. A tensor is
2:29nothing but a container for numbers.
2:31Sometimes you store characters or
2:33strings too. But that is very rare.
2:35Almost never do you do that. 99.99%of
2:38the time, it is going to be a container
2:41for numbers. Right? Now guess what? You
2:44have used tensors before. I mean, even
2:47if you haven't heard the name, you have
2:48used them. So if you have ever studied
2:51about vectors or matrices in your life,
2:53they are actually tensors. So when it
2:56is zero-dimensional, you call it a
2:58scalar. A single number. When there is
3:01a list of numbers, you call it a vector
3:03. When there are numbers in 2D, you
3:06call it a matrix. Right? Now think, if
3:09there are numbers in 3D, what would you
3:10call it? Or what would you call it in
3:124D? So that is why computer scientists,
3:15in fact, this is an area combining
3:17physics and math. They decided.
3:20Scientists decided to coin a general
3:22term. And they use, they call it a
3:25tensor. Right? So let's study more in
3:27detail about tensors. I will share my
3:29screen. So yeah, tensors are basically
3:33a container for storing numbers. We
3:37have discussed this. Let's talk about
3:390D tensors, which are scalars. Right?
3:43So whenever you are storing a single
3:45number somewhere like this or like this
3:48. These are basically called scalars.
3:52Right? Or you can call it a 0D tensor.
3:58Right? 0D means that the number of
4:02dimensions of this particular tensor is
4:06zero. There are no dimensions. Right?
4:10If I show you practically, what I can
4:13do is I can go to Google Colab. If you
4:16don't know Google Colab, trust me, it's
4:19a very simple tool where you can run
4:22Python code online. Right? You don't
4:24need to know more than that. I am
4:26importing a library. Called NumPy,
4:28which we will use a lot in this video,
4:30and in machine learning in general,
4:31this library is used heavily. If you
4:33want to study NumPy, I will put a link
4:36to a playlist from my own channel in
4:38the description of this video. You can
4:40study NumPy from there. NumPy is very
4:42simple. All right? Now, if you want to
4:45create a scalar using NumPy. Then you
4:47don't have to write much. You just need
4:48to write one line of code. Let's say we
4:50are creating a variable A in which we
4:51will store the scalar. This will be
4:54equal to np.array and here I put a
4:58number and ran it. And as soon as I
5:03printed A, which is on your screen,
5:06this is actually a scalar or a 0D
5:09tensor. All right? How do we know it's
5:120D? Whenever you are creating tensors
5:14in NumPy. See, one more thing to
5:16remember, if you are from a computer
5:18science background, then a tensor and
5:20an n-dimensional array are the same
5:22thing. An array means one-dimensional.
5:24A 2D array means an array within an
5:26array. A 3D array means an array within
5:27an array within an array. All right? So
5:29in that way, n-dimensional array is the
5:31general term if you come from a math or
5:33physics background, it's called a
5:35tensor. So the machine learning
5:36community has adopted that term. So
5:37whenever you see an n-d array, just
5:40understand that it is a tensor. All
5:42right? So here you have a 0D array.
5:44Because there is no array at all. There
5:45is just one number. So you call this a
5:470D tensor. So whenever you create
5:49arrays or tensors in NumPy, you can
5:52find their dimension. You have an
5:54attribute called ndim. It's clear from
5:57the name. Number of dimensions. As soon
5:59as you run this, you get zero. From
6:02this, you understand that the dimension
6:03of this particular tensor is zero. And
6:06that is why it is a 0D tensor. All
6:08right? So I showed you a small
6:10demonstration of this. Let's quickly
6:12discuss a 1D tensor. So if you have a
6:15list of numbers like this 1, 2, 3, 4,
6:20then this is a 1D tensor, and you can
6:25also call it a vector, and in
6:29programming, you can call it a 1D array
6:35or simply an array. All right? Now, for
6:39this thing, this tensor, what will its
6:42ndim be? It would be one. Meaning, this
6:46is a tensor that has one dimension. All
6:48right? Now you need to understand one
6:51more thing, what is an axis in terms of
6:53tensors? So an axis is basically how
6:56many dimensions you have. All right?
6:58Basically, if you have two dimensions.
7:01Let's say your tensor has two
7:02dimensions. We will discuss 2D tensors
7:04a little later. So this means there are
7:06two axes there. Okay? If it is a 3D
7:09tensor, then you have three axes. Okay?
7:11So dimensions and axes are related.
7:15Okay? Whatever the D of the tensor is,
7:17whatever the dimension of the tensor is
7:19, you will have that many axes. And one
7:22more thing, the number of axes is also
7:25called rank in some places, and you
7:28also call that dimension. Okay? So,
7:32there shouldn't be any confusion here.
7:34If you go and read in different places,
7:35you will see different things. I am
7:37just letting you know. Number of axes
7:39is equal to rank, equal to the
7:40dimension of the tensor. Okay? Let me
7:43quickly show you how you can create a
7:461D tensor using NumPy. So, I am
7:49creating an array. And here, what I am
7:52doing is sending a Python list inside
7:54the array that contains these values.
7:58And now if I print the array, this,
8:00which is on your screen, is a 1D tensor
8:03. Okay? And if I write arr.ndim, you
8:07would see that its n-dimension is one.
8:11So, the number of axes inside it is one
8:13. Its rank is also one, and it is a 1D
8:16tensor. Okay? I guess you have no
8:18confusion up to this point. Now I will
8:19say something that might confuse you a
8:21little. But it is very important to
8:22understand. Okay? Listen carefully.
8:25This 1D tensor, you also call this a
8:32vector. This is also a vector. You can
8:35call it a vector. Okay? But if someone
8:37asks you what the dimension of this
8:39vector is? Then, since there are four
8:42numbers here, you would say the
8:44dimension of this vector is 4D. Now you
8:47will say that I just said that this is
8:491D. I said it is a 1D tensor. This is a
8:521D tensor. And since it is a 1D tensor,
8:55it means it is a vector. But how many
8:57dimensions are inside this vector? Four
8:59dimensions. Now this might seem a
9:01little confusing to you. But I will
9:03give you an example in a little while
9:04that will make this crystal clear as to
9:06what I am trying to say. At this point,
9:10just create a distinction in your mind
9:12that whenever you create a 1D tensor,
9:14it is a vector, and the number of
9:16values in that vector determines the
9:18dimension of that vector. Meaning, if I
9:22give you another example, let's say one
9:25and two, this is still a 1D tensor. And
9:29it is also a vector. But if someone
9:31asks you, how many dimensions does this
9:32vector have? Then you would say two,
9:34because there are two numbers in it. I
9:35hope you guys are following along so
9:38far. Okay? If you are following along
9:41so far, you must have also noticed one
9:43thing: if you keep adding more scalars
9:46to a scalar, then it becomes a vector.
9:52Meaning, if you added four scalars here
9:56, you got a vector. Always remember
9:59this. In the case of tensors, as you
10:01increase the dimension, how do those
10:03dimensions always increase? If you keep
10:06adding the previous dimension, you get
10:08the next dimension. Meaning, if you
10:11want a 1D tensor, basically it is a
10:13collection of multiple 0D tensors. If
10:17you want a vector, it is basically a
10:18collection of scalars. If you want
10:21matrices, it is basically a collection
10:23of vectors. So, you must always
10:24remember this fundamental. Okay? Now
10:27let me show you an example of a 2D
10:28tensor, which we also call matrices.
10:30Okay? So now let's discuss what are 2D
10:33tensors or matrices. So, imagine if you
10:37have multiple vectors, like. So If you
10:47add or collect these vectors, then we
10:52call it a matrix. Okay? Something like
10:55this. 1 2 3 4 5 6 7 8 9, so basically
11:04matrices are nothing but a collection
11:07of vectors. Okay? And obviously, its
11:11dimension is 2D because there are two
11:14axes here. Okay? This one axis is the
11:16column axis, and this one axis is the
11:18row axis. Okay? So, I hope you are
11:20understanding this. Its rank is two,
11:23which is equal to the dimension, and
11:26there are two axes: rows and columns.
11:29Let me quickly show you how this is
11:31created. It is very simple. You will
11:34use NumPy again. Let's say I am
11:36creating a variable called MAT, and MAT
11:40is equal to np.array, and here what
11:43will I do? I will pass a list of lists
11:45in Python, something like this. 1, 2, 3
11:47, then a comma; the second item itself
11:50is going to be a list 4, 5, 6. And 7, 8
11:56, 9, okay. And if we print the MAT, you
12:00will get a structure like this; this is
12:02a matrix, okay. If you calculate its MM
12:06, then as expected, it would be two. 2D
12:13Tensor. Okay, now you can easily
12:15understand that if you want to create
12:17higher dimension tensors from this,
12:19what will you do? Basically, if you
12:22want to create a 4D tensor. Sorry, if
12:25you want to create a 3D tensor, you
12:27will have a matrix that looks like this
12:30. You will take more matrices like this
12:35. This is one matrix. This is the
12:38second one. This is the third one. So,
12:42basically, suppose if all these
12:44matrices have 3 by 3 numbers and you
12:47are taking four such matrices, then you
12:51will get a 4 by 3 by 3 tensor, which
12:54you will call a 3D tensor, and this
12:57time there are three axes here. Okay?
13:01So you have a column axis. You have a
13:04row axis and you have a depth axis.
13:08Okay? This is a 3D tensor. Okay? Or you
13:11can actually see it in terms of a
13:13cuboid. Right? Something like this.
13:19Right? Now, if someone asks you what a
13:224D tensor would look like? What would a
13:244D tensor be like? So a 4D tensor is
13:26basically a collection of 3D tensors.
13:28If this is a 3D tensor, then a 4D
13:30tensor will look like this. Basically,
13:36it will be a vector of 3D tensors. See
13:38like this. Now, sorry for my drawing.
13:42But these lines have 3D tensors placed
13:45in this way. These are what you call a
13:484D tensor. Right? Now, if someone asks
13:51you what a 5D tensor would look like?
13:53So a 5D tensor will be a matrix of 3D
13:55tensors. Basically, more cuboids will
13:59come underneath it like this. Like,
14:02this is a 5D tensor. Right? And that is
14:08how this entire thing works. This is a
14:115D tensor. I mean, if you focus only on
14:13this much. If you focus only on this
14:16much, then this is a 4D tensor. But if
14:22you focus on the whole thing. Focus on
14:24this whole thing, this is a 5D tensor.
14:26Okay? And that is how this entire thing
14:29is stacked on top of each other. Now,
14:30if you make a collection of 5D, it will
14:33become 6D. If you make a collection of
14:356D, it will become 7D. That is the
14:36whole idea. So where you were
14:38restricted to thinking only about
14:40scalars, vectors, and matrices, now you
14:42are completely free. Because you have a
14:45general concept that we call this thing
14:47a tensor, and in terms of that tensor,
14:49you can think or do things in any
14:51number of dimensions. Okay? Now, if you
14:54talk about machine learning, you mostly
14:56revolve between zero to 5D. You won't
14:59really find tensors with higher
15:00dimensions than that. Okay? So that’s
15:02why, in today’s video, I will show
15:04you one example of each of the five
15:06tensors. Okay? To be honest, there’s
15:09no need to show an example of a scalar.
15:11So I will show you all examples from 1D
15:13up to 5D. Practical examples, the kind
15:15of examples you might encounter later
15:16on. Okay? But before we go there, I
15:18just want to tell you one more thing.
15:20This is the last concept. Although we
15:22have discussed it a little bit already.
15:23What do we call rank, axis, and shape?
15:25Okay? Let’s discuss it quickly. Rank
15:28is just that. The number of dimensions
15:30your tensor has is what is called its
15:32rank. It is also called the number of
15:35axes. Number of axes is equal to rank,
15:40is equal to the number of dimensions.
15:45These three things are the same, guys.
15:46Okay? You must always remember this.
15:48Let me explain it to you in terms of a
15:51matrix. Suppose you have this matrix.
15:53It is a 3x3 matrix. So, obviously, it
15:56has two axes. One row axis and one
15:59column axis. One more thing comes up
16:02here. Then what is shape? Shape is how
16:07many items you have in any particular
16:10axis. For example, if you talk about
16:13the row axis, how many maximum items
16:15can you store in the row axis? Three.
16:18Right? It’s 3x3, so you can only
16:20store three. So the shape of this
16:23matrix is 3x3, but you can also have a
16:25matrix like this. This is also a matrix
16:30. If you calculate its ndim or its rank
16:34, it will still be two. Because it
16:38still has a column axis and a row axis
16:40here. But this time, if you calculate
16:43the shape, it will be 2x3. Why am I
16:46saying that? Because in the row, in the
16:48row axis, how many maximum items can
16:50you store? Two. And how many can you
16:52store in the column axis? Three. Okay?
16:55Let’s take one more example. This is
17:004x2. Okay? There is one more thing,
17:03size. Size of a tensor. So size means
17:06how many items are in that tensor. So
17:08simply multiply its shape, all the
17:10numbers. Multiply these. 2x3 is six.
17:14Meaning there are six items inside it.
17:16If you multiply 3 here, you get nine.
17:18Meaning there are nine items here. Here
17:20, multiply 4 by 2, you get eight.
17:21Meaning there are eight items. Okay? So
17:23, whenever you want to find out how
17:25many items are in a particular tensor,
17:27which we call size, it is very simple
17:28to calculate. You just find its shape
17:30and multiply all the numbers. Okay?
17:32This is just not true for scalars. For
17:34a scalar, the size is always equal to
17:37one. It's simple. If it is a scalar,
17:39its size will be one. For tensors above
17:41that, you simply have to multiply their
17:43shape. If you have a vector, a 1D
17:45tensor, it will look something like
17:48this. So in this case, your shape will
17:52be three. Okay? Because here, in one
17:55single place, there are three items.
17:57Okay? So here, three is your number of
18:00items. Okay? I hope you are
18:02understanding. It is very simple. Once
18:04you grasp the concepts. Okay? So rank,
18:06axis, and shape, these are the three
18:08things we discussed. In fact, we also
18:10discussed size. Now, the most important
18:13part of this video: we will see
18:14practical examples of tensors from 1D
18:17to 5D that you will encounter later in
18:19your life while doing machine learning
18:21and deep learning projects. We are
18:23going to do that next. Let me now give
18:26you two practical examples of a 1D
18:27tensor that you will see everywhere in
18:29machine learning in the future. Okay?
18:32And this is where you will understand
18:35why I said a little while ago that a
18:37vector is a 1D tensor but its own
18:39dimensions are different. You will
18:42understand this now. Okay? And to
18:44explain this whole thing, I will take
18:45an example. Let's say we have a dataset
18:48about students. We have discussed this
18:50dataset before as well. We have a
18:52dataset of students. And in that
18:55dataset, we have four columns. The
18:57first column is CGPA. The second column
19:02is IQ. The third column is state. Which
19:05state are they from? West Bengal,
19:07Kerala, or whatever it is. Okay? And
19:09here it is whether they got placed or
19:12not. Which is your target column that
19:14you need to predict. Classification
19:16problem. Okay? Now, let's assume I have
19:20data for 10,000 students, and obviously
19:22, my end goal is to build a
19:25classification model, so if I provide a
19:28new student's details—meaning IQ,
19:30CGPA, and state—it should tell me
19:33whether they will be placed or not. I
19:36need to make this prediction. Right? So
19:38, what do I do here? I will show you an
19:41example of a 1D tensor. I'll show you
19:44an example of a 1D tensor or a vector.
19:47For that, you'll have to forget the
19:4910,000 students and focus on just one
19:52student. If you focus only on the first
19:54student. Let's say our first student
19:57has a CGPA of 8.1, an IQ of 91, is from
20:01West Bengal, and they got placed. For
20:05now, we are ignoring the placement
20:07column because we are going to focus on
20:09the input columns. So, if you take
20:12their input, if you take that student's
20:14input, basically it is a set of three
20:16things. Now, let's assume for a moment
20:19that our data only contains students
20:21from two states. The first state is
20:23West Bengal, the second is Bengaluru,
20:26Karnataka. Right? And assume we are
20:29calling West Bengal zero and Karnataka
20:32one. We have performed numerical
20:34encoding. This is called label encoding
20:35. You will understand this in future
20:37videos. So, basically, instead of West
20:40Bengal, I wrote zero. So if we only
20:43talk about the first student, we have
20:46these numbers: 8.1, 91, and 0, and
20:49guess what? This is a tensor, a 1D
20:52tensor. Or you could call it a vector.
20:56This is a vector. And what is the
20:58dimension of this tensor? 1D, because
21:00it has only one axis. There is only one
21:03axis. But what is the dimension of this
21:05vector? It is a 3D vector. Why am I
21:08saying that? Because there are three
21:11axes here. There are three axes here.
21:13What is the first axis? CGPA. What is
21:16the second axis? IQ, and what is the
21:19third axis? State. And this student is
21:24somewhere here. There is that vector.
21:27This is the vector. Now, suppose if
21:31there were another student, 7.2 IQ, 102
21:33, Karnataka, no placement, so assume
21:36they would lie somewhere here,
21:38something like this. Right? They would
21:42have their own separate vector. So,
21:45basically, if you talk about it, in
21:47this three-dimensional space, you have
21:49data for 10,000 students. Data for
21:5110,000 students. Meaning you have
21:5410,000 vectors. Each vector represents
21:56a student in this three-dimensional
21:59coordinate space. So when you are
22:01talking about vectors, this is a 3D
22:02space. But when you represent those
22:05numbers in the form of a tensor, it is
22:07a 1D tensor. So these are two different
22:09things. Do not get confused between
22:11them. Are you talking about the
22:13dimension of the tensor or the
22:15dimension of the vector? There is a
22:17difference in that. Right? It became 3D
22:19here because you had three input
22:21columns. If you have 50 input columns,
22:24then you are working in a 50-
22:26dimensional coordinate space. But it
22:28will still remain a 1D tensor. Please,
22:31I hope you are understanding this point
22:33. Okay? One more example where you will
22:36find a 1D tensor. One example is
22:37already this. Whenever you get any
22:39machine learning tabular data, each row
22:42is actually a 1D tensor or a vector.
22:45Okay? One more example is, now think if
22:47you had data for these 10,000 students
22:49here, then everyone's output would look
22:52something like this. In this way. Now
22:54if you take out all these numbers. If
22:57you extract all these numbers and
22:59represent them in this manner as well.
23:06Then there will be a total of 10,000
23:09numbers in it. But this is also a 1D
23:13tensor. This is also a 1D tensor. Okay?
23:18So I hope you are understanding a bit
23:20by now where you can find 1D tensors in
23:22machine learning. Okay? Now let me show
23:25you where you will find 2D tensors.
23:28Where will you find 2D tensors? It is a
23:30very simple thing, guys. What is a 2D
23:32tensor? A 2D tensor is basically a
23:34collection of 1D tensors. Right? So if
23:37you have the data we had earlier of
23:41CGPA, IQ, state, and placement. Now
23:45let's forget about placement for a
23:47moment. If we only focus on the input
23:48columns. We had data for 10,000
23:50students. Like this. So here, every
23:55student's data is actually a vector or
23:59a 1D tensor. So a collection of 10,000
24:02vectors would be what? A matrix, right?
24:06So you can actually write this entire
24:08thing inside a matrix. So whenever you
24:12get data in machine learning, its input
24:15—the collection of input columns—
24:18you can store all that data in a matrix
24:21. That is generally called a 2D tensor
24:23in machine learning. It is denoted by
24:25D. Okay? So it will look something like
24:28this. Data of the first student, data
24:31of the second student, data of the
24:34third student, data of 10,000 students.
24:38And this becomes your 2D tensor. Okay?
24:43Which is a collection of 10,000 vectors
24:46that are themselves 3D vectors. I hope
24:48you are understanding this so far. And
24:51this placement, whether it happened or
24:54not, is itself a 10,000-dimensional
24:57vector. And basically, it is a 1D
25:00tensor. This is a 1D tensor. Okay? This
25:03is a 2D tensor. I hope you understood
25:06this entire discussion. You will
25:08encounter this every time. If you ever
25:10work with tabular data, you are working
25:13with 1D tensors as well as 2D tensors.
25:16Basically, you are working with
25:18matrices and also with vectors. Or in
25:20summary, if I tell you, when you
25:23separate all the inputs, you can call
25:25it a matrix or a 2D tensor. And when
25:28you isolate the entire input, which is
25:30a single column, it becomes your vector
25:31or your 1D tensor. Okay? So, I guess
25:34you understand where you find 1D and 2D
25:36tensors. Now let's move on to 3D
25:38tensors. So now let's focus our
25:40attention on 3D tensors. To be honest,
25:423D tensors are a bit rare. Calling them
25:45rare would actually be wrong. It's just
25:47that they aren't as frequent as your 1D
25:50and 2D tensors. If you want to see a
25:52practical example of 3D tensors. In
25:53fact, I will show you two practical
25:55examples. One of them, a very intuitive
25:57and interesting practical example, is
26:00from the domain of NLP, Natural
26:01Language Processing. Meaning, if you
26:04are ever working with any textual data,
26:07you will see 3D tensors there. Let me
26:09show you how. So let's say I have some
26:12text written. Hi Nitish. Hi Rahul and
26:20Hi Ankit. These are three texts. And
26:24assume these three texts are my input.
26:27I have to give these three texts to the
26:29algorithm to train. Now, obviously, you
26:32know that machine learning algorithms
26:33are pure mathematics. They don't
26:35understand text or strings. So you will
26:38have to convert this text into numbers,
26:41or, in other words, into vectors. This
26:43entire process is actually called
26:45vectorization in the NLP domain. Where
26:47you convert a given text into numbers
26:50or vectors. So that is a very important
26:52thing. And there are many different
26:54vectorization techniques. But we are
26:55going to focus on a very simple and
26:57very intuitive technique. Let me show
26:59you. So what I will do is, I need to
27:02convert all these sentences into
27:04numbers. For that, what I will do is, I
27:08will list down all my words, all my
27:11unique words. Like 'Hi' once, 'Nitish'
27:14once, basically I wrote down my entire
27:17vocabulary. Vocabulary means the set of
27:20all unique words present in my corpus,
27:22as it is called here. Basically, in my
27:25text. Okay? Now what will we do? We
27:29will represent every word, every
27:31particular word, with a vector. For
27:34instance, if I have to represent 'Hi',
27:37then 'Hi' becomes 1 0 0 0. Whenever I
27:42find 'Hi' written anywhere, I can write
27:44this instead. If I find 'Nitish'
27:46written anywhere, I will write 0 1 0 0;
27:50that becomes 'Nitish'. Okay? If I have
27:53to represent 'Rahul', then it becomes
27:55this. And if I have to represent 'Ankit
27:58', then it becomes this. Right? Now, if
28:01I talk about these sentences. If I talk
28:03about these sentences, let's say my
28:05first sentence is what? 'Hi Nitish'. So
28:07what is this actually? It is a set or
28:10collection of two vectors. So the first
28:14vector is 1 0 0 0 and the second vector
28:18is 0 1 0 0; this is my first sentence.
28:24If you notice carefully, then the inner
28:27vector is a 1D tensor, and there are
28:30two such 1D tensors. So it means this
28:32entire thing, the outer one, is a 2D
28:34tensor. Right? Now, if the first
28:37sentence is a 2D tensor. Then the
28:40second sentence will also be a 2D
28:41tensor. And the third sentence will
28:43also be a 2D tensor. Basically, you
28:46have a collection of three 2D tensors.
28:49So what has it basically become? A 3D
28:51tensor. Right? So if I want to write it
28:53, I will write it here again. This is
29:02my second 2D tensor and this is my
29:06third sentence. Right? And if I want to
29:11write their collection, then this
29:15became my 3D tensor. Right? If I have
29:19to write its shape, how would the shape
29:21be? We have vectors of four. Like,
29:27vectors of size four. And the matrices
29:30I have are of size 2/4. Okay? Meaning,
29:35basically two rows, four columns, and I
29:38have three such matrices. So this is
29:42the shape, guys, of my 3D tensor. There
29:45are three axes here. Meaning, it is
29:47rank three. And how many items are
29:48there? 3*2*4, 24 items, which is
29:52actually correct. Okay? So, this is one
29:56good example of a 3D tensor. Okay? I
29:59will give you one more example of a 3D
30:01tensor. Okay? Uh, watch carefully. So,
30:13another good example of a 3D tensor is
30:15time series data. Time series data
30:23means data that you collect after a
30:27frequent time period. For example,
30:30share market data that you collect on a
30:32daily basis, or sometimes you collect
30:35every hour. Depends. Okay? I will show
30:37you one example. Let's say, what data
30:39are you storing? The highest and lowest
30:44price of any stock during the day. Okay
30:51? And you are filling this data once a
30:54day for this entire stock, and you are
30:57doing this for the whole year. Okay? So
31:01the data you will have will be 365 by 2
31:04, think about it, because it will look
31:07like this: Day one, some number, some
31:10number; Day two, some number, some
31:13number; Day 365, some number, some
31:16number. Right? So basically, you have
31:19365 on the row axis and two on the
31:21column axis, because you have two input
31:23columns. So this data you have is a 2D
31:28tensor. If you are tracking two
31:30quantities throughout the day for the
31:32whole year, then it is a 2D tensor. But
31:35what if I have 10 years of data for
31:38this same stock? I have tracked this
31:41thing for 10 years. So, how many such
31:442D tensors do I have? 10, so the
31:48resultant tensor will be 10 cross 365
31:52cross 2, which is a 3D tensor. A
31:58collection of 10 2D tensors is a 3D
32:01tensor. Right? So, whenever you get
32:04time series data, it is 3D data. And
32:06the middle axis in it, this axis here,
32:09is your time axis. This is your time
32:12axis. Okay? So you will work a lot on
32:14time series later on. Most financial
32:16data, or often medical data, is time
32:18series-based data, and a lot of
32:20analysis is done on it, and it is a
32:23very important thing, and this is an
32:25example where 3D tensors are used. Okay
32:28, I hope you are understanding up to
32:29this point. Now we have 4D tensors and
32:315D tensors left, let's complete them,
32:33let's cover them. Now let's focus on 4D
32:35tensors. Okay, a great example of a 4D
32:38tensor is images, right? Image-based
32:42data. So, if you ever work in the
32:46domain of computer vision, or work with
32:49images, then the tensors you encounter
32:52there are 4D tensors. Let me show you
32:55how. Okay? If you have ever studied
32:58even a little bit of image processing,
33:01you might know that any image is
33:03basically a collection of pixels.
33:06Pixels look like this. There are very
33:09small pixels on your screen like this.
33:12And every pixel has a numerical value
33:15because of which you see something.
33:18Like, if this value is black, this
33:20value is black, and all these values
33:23are black, then you see a two on your
33:25screen. Everything else was one. Only
33:28these values are black. So this is a
33:31two on your screen. So by controlling
33:33these pixels, you can display any image
33:36. Now, if you want to show a color
33:39image, you take three channels like
33:42this. This is the first channel that I
33:45showed you. You take three channels
33:47like this. This is R, this is G, and
33:49this is B. Red, Green, Blue. Okay? And
33:52you stack these three on top of each
33:53other like this. Right? So suppose
33:56these three are obviously matrices.
34:01These three are actually 2D tensors.
34:03And if you combine three 2D tensors, it
34:05becomes a 3D tensor. Right? Suppose one
34:09has a shape of 1200 by 800, and you
34:12have three such channels, so combining
34:17them you get 3, 1200, 800—three
34:20channels and 1200*800 pixels in each
34:24channel, okay? So that's for one image.
34:28If you are talking about that, it is a
34:313D tensor, right? But what if you have
34:34multiple images? Suppose you have 50
34:37color photographs like this. If you
34:40have 50 color images. In that case, you
34:45get a 50 by 3 by 1200 by 800 tensor.
34:51And this is your 4D tensor. So in the
34:55future, if you ever do image processing
34:57or study convolutional neural networks
34:59in deep learning, when you work with
35:01images there, you work with batches of
35:03images. And those batches of images are
35:06a great example of a 4D tensor. Okay?
35:09So I hope you understood this example
35:11as well. Now let's move on to 5D
35:13tensors. So I guess you must have
35:15already understood 5D tensors. A very
35:17good example of that is videos. Right?
35:22Because what are videos? If you think
35:24carefully. You probably already know.
35:26Videos are basically images that are
35:29passed in front of your eyes very
35:31quickly. Because they pass at such a
35:35speed, our eyes and our brain cannot
35:37distinguish that they are two separate
35:40images. Because the speed at which the
35:44images are crossed is higher than our
35:47persistence of vision. So I will show
35:50you, it is written here that our
35:53persistence of vision can only classify
35:5512 separate images in one second. If
35:58you pass more than 12 images in front
36:01of our eyes in one second. Then it
36:03gives us a feeling of a video. Right?
36:06And this generally happens with 30fps,
36:0860fps, 120fps; I am not sure about
36:10120fps. But videos are passed at up to
36:1460 frames per second. So obviously
36:16those images are passing so fast that
36:18we feel it is in continuity. It is a
36:20video. Right? So in short, videos are
36:25basically frames. Right? A collection
36:30of frames. Right? A single frame means
36:32one image. One frame means one image.
36:35Right? So let's say you have a 60-
36:41second video that was shot at 30 FPS
36:49and is 480p. 480p means its resolution
36:54is 480 by 720. Obviously, this is color
36:58, so it will have three channels. RGB.
37:01Right. So now look, a single image
37:04inside it is obviously 480. 720, 3, a
37:09single image, right. Now I have a 60-
37:14second video, shot at 30 FPS, so how
37:17many images do you have? If you want to
37:21calculate this, you are getting 30
37:23videos in one second, 30 frames per
37:25second. In one second you are getting
37:2830 videos, sorry, 30 images. And you
37:30have 60 seconds like that. So that
37:32means you have 1800 images. Right? If
37:36there are 30 in 1 second, then in 60...
37:39sorry, 1800, please ignore if there is
37:41any calculation mistake. You have 1800
37:45images. Right? This is already your 4D
37:48data. Now suppose you have four video
37:50clips like this. You have got four
37:53videos. So now the data you have, this
37:56collection of four videos, is actually
37:58a 5D tensor. This is a 5D tensor. The
38:03collection of videos is a 5D tensor.
38:05Where the first thing tells you how
38:07many videos there are. This part tells
38:09you how many images are in a single
38:11video, and this is the size of each
38:13image and how many color channels it
38:14has. So, this is a 5D tensor. Now, if
38:17you multiply these numbers. If you
38:19multiply these numbers, it becomes a
38:21very large number. Right? Let me show
38:23you how large this number will be. This
38:27number will be: we have four videos
38:30multiplied by 1800 images, multiplied
38:34by 480 by 720. I want to show you the
38:38calculation I am doing. Multiplied by 3
38:42, this is the number, guys. This is the
38:44number and this is a big number. Right?
38:47Now think, if you have this many values
38:49, basically there are this many numbers
38:51here. Right? Because if you multiply
38:54all these, the tensor has that many
38:55values. That many items. So, there are
38:58this many items here. Now think, if you
39:01store each item in a float, in a
39:04float32. Basically, if you store every
39:07single item in a 32-bit float, how much
39:10total storage will you need? Multiply
39:14by 32, meaning you would need this many
39:17bits. So, how much would this number be
39:19in bytes? Divided by 8, you would need
39:22this many bytes. If I want to convert
39:25this into kilobytes, I will divide it
39:28by 1024. Here is the number in
39:31kilobytes. If I want to convert it into
39:34megabytes, I will divide it again by
39:371024. Such a huge megabyte. And if I
39:41divide it again by 1024, here is our GB
39:45, 27 GB. Meaning, if you want to store
39:50four 60-second videos, or process them,
39:54or run machine learning or deep
39:57learning on them, you would need 27 GB
40:01of space. Obviously, you don't do that,
40:06and that is where video codec formats
40:10come in, video encoding formats like
40:13MKV or MP4. Okay, what do they do?
40:17Whatever you are giving to a single
40:19item, they reduce it significantly, and
40:22you can then do all this work with much
40:25less data. Honestly speaking, video
40:27processing is somewhat rare, unless you
40:29are building an application like
40:30YouTube where you have to process every
40:32video before it gets uploaded. You most
40:34likely won't be working on videos, but
40:36you might if you are working in the
40:38computer vision domain. You never know,
40:40right? But yeah, you personally won't
40:42have sensors beyond 5D, there aren't
40:44really any use cases beyond that. Okay,
40:47so 5D will be the max and that's it.
40:50That's what I wanted to discuss in this
40:51video. I know it has become a quite
40:53long video, it must be around 40
40:55minutes, I guess. So what can I say, I
40:58taught it in a bit of detail. I hope
41:01you understood it and this will provide
41:04us a good start when we do things ahead
41:07. Okay, so yeah, that's it guys. If you
41:10liked the video, please consider
41:12subscribing. If possible, do share it
41:14with someone if they are learning
41:15machine learning. And let's meet
41:17tomorrow, tomorrow we will do an
41:19end-to-end machine learning example so
41:21that you get a bird's eye overview of
41:23the whole process. So yeah, thanks for
41:25watching, will meet tomorrow, bye.

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