Imagine your have five different classes e.g. ['cat', 'dog', 'fish', 'bird', 'ant']. If you would use one-hot-encoding you would represent the presence of 'dog' in a five-dimensional binary vector like [0,1,0,0,0]. If you would use multi-hot-encoding you would first label-encode your classes, thus having only a single number which represents the presence of a class (e.g. 1 for 'dog') and then convert the numerical labels to binary vectors of size .

Examples:

'cat'  = [0,0,0]  
'dog'  = [0,0,1]  
'fish' = [0,1,0]  
'bird' = [0,1,1]  
'ant'  = [1,0,0]   

This representation is basically the middle way between label-encoding, where you introduce false class relationships (0 < 1 < 2 < ... < 4, thus 'cat' < 'dog' < ... < 'ant') but only need a single value to represent class presence and one-hot-encoding, where you need a vector of size (which can be huge!) to represent all classes but have no false relationships.

Note: multi-hot-encoding introduces false additive relationships, e.g. [0,0,1] + [0,1,0] = [0,1,1] that is 'dog' + 'fish' = 'bird'. That is the price you pay for the reduced representation.

Ideally, you could train you model to classify input instances and producing a single output. Something like

y=1 means input=dog, y=2 means input=airplane. An approach like that, however, brings a lot of problems:

1. How do I interpret the output y=1.5?
2. Why I'm trying the regress a number like I'm working with continuous data while I'm, in reality, working with discrete data?

In fact, what are you doing is treating a multi-class classification problem like a regression problem. This is locally wrong (unless you're doing binary classification, in that case, a positive and a negative output are everything you need).

To avoid these (and other) issues, we use a final layer of neurons and we associate an high-activation to the right class.

The one-hot encoding represents the fact that you want to force your network to have a single high-activation output when a certain input is present.

This, every input=dog will have 1, 0, 0 as output and so on.

In this way, you're correctly treating a discrete classification problem, producing a discrete output and well interpretable (in fact you'll always extract the output neuron with the highest activation using tf.argmax, even though your network hasn't learned to produce the perfect one-hot encoding you'll be able to extract without doubt the most likely correct output )

You can think of binary encoding as a compromise between label encoding and one-hot encoding. For distinct categories, label encoding introduces a false linear order that brings a lot of noise into the model (category 1 < category 2 < category 3....) . Binary encoding introduces false additive relationships between the categories (e.g. category 4 + category 1 = category 5 or 100 + 001 = 101) but fewer of them.

Therefore, binary will usually work better than label encoding, however only one-hot encoding will usually preserve the full information in the data.

Unless your algorithm (or computing power) is limited in the number of categories it can handle, one-hot encoding will be preferred over other encoding schemes. If you are limited, mean encoding is a powerful alternative because it transform a categorical feature into a numeric one (giving you the minimal number of inputs) while preserving the most important information in the data.

[Mean encoding replaces every category with its target mean. The mean encodings need to be constructed carefully on a separate dataset to avoid data leakage. If you want to reduce your input dimensions as much as possible, this can be helpful. Mean encoding can also be helpful as an additional feature and is very popular on Kaggle to squeeze out some extra performance. This is also known as target encoding or likelihood encoding.]