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HomeArtificial IntelligencePosit AI Weblog: Collaborative filtering with embeddings

Posit AI Weblog: Collaborative filtering with embeddings


What’s your first affiliation once you learn the phrase embeddings? For many of us, the reply will most likely be phrase embeddings, or phrase vectors. A fast seek for latest papers on arxiv reveals what else could be embedded: equations(Krstovski and Blei 2018), automobile sensor knowledge(Hallac et al. 2018), graphs(Ahmed et al. 2018), code(Alon et al. 2018), spatial knowledge(Jean et al. 2018), organic entities(Zohra Smaili, Gao, and Hoehndorf 2018) … – and what not.

What’s so enticing about this idea? Embeddings incorporate the idea of distributed representations, an encoding of data not at specialised places (devoted neurons, say), however as a sample of activations unfold out over a community.
No higher supply to quote than Geoffrey Hinton, who performed an necessary position within the growth of the idea(Rumelhart, McClelland, and PDP Analysis Group 1986):

Distributed illustration means a many to many relationship between two kinds of illustration (equivalent to ideas and neurons).
Every idea is represented by many neurons. Every neuron participates within the illustration of many ideas.

The benefits are manifold. Maybe essentially the most well-known impact of utilizing embeddings is that we are able to be taught and make use of semantic similarity.

Let’s take a job like sentiment evaluation. Initially, what we feed the community are sequences of phrases, basically encoded as components. On this setup, all phrases are equidistant: Orange is as completely different from kiwi as it’s from thunderstorm. An ensuing embedding layer then maps these representations to dense vectors of floating level numbers, which could be checked for mutual similarity through numerous similarity measures equivalent to cosine distance.

We hope that once we feed these “significant” vectors to the following layer(s), higher classification will end result.
As well as, we could also be all in favour of exploring that semantic area for its personal sake, or use it in multi-modal switch studying (Frome et al. 2013).

On this submit, we’d love to do two issues: First, we need to present an attention-grabbing utility of embeddings past pure language processing, particularly, their use in collaborative filtering. On this, we observe concepts developed in lesson5-movielens.ipynb which is a part of quick.ai’s Deep Studying for Coders class.
Second, to collect extra instinct, we’d like to have a look “underneath the hood” at how a easy embedding layer could be carried out.

So first, let’s bounce into collaborative filtering. Similar to the pocket book that impressed us, we’ll predict film scores. We’ll use the 2016 ml-latest-small dataset from MovieLens that incorporates ~100000 scores of ~9900 films, rated by ~700 customers.

Embeddings for collaborative filtering

In collaborative filtering, we attempt to generate suggestions primarily based not on elaborate data about our customers and never on detailed profiles of our merchandise, however on how customers and merchandise go collectively. Is product (mathbf{p}) a match for person (mathbf{u})? If that’s the case, we’ll advocate it.

Usually, that is executed through matrix factorization. See, for instance, this good article by the winners of the 2009 Netflix prize, introducing the why and the way of matrix factorization methods as utilized in collaborative filtering.

Right here’s the overall precept. Whereas different methods like non-negative matrix factorization could also be extra widespread, this diagram of singular worth decomposition (SVD) discovered on Fb Analysis is especially instructive.

Figure from https://research.fb.com/fast-randomized-svd/

The diagram takes its instance from the context of textual content evaluation, assuming a co-occurrence matrix of hashtags and customers ((mathbf{A})).
As said above, we’ll as an alternative work with a dataset of film scores.

Have been we doing matrix factorization, we would wish to someway handle the truth that not each person has rated each film. As we’ll be utilizing embeddings as an alternative, we received’t have that drawback. For the sake of argumentation, although, let’s assume for a second the scores had been a matrix, not a dataframe in tidy format.

In that case, (mathbf{A}) would retailer the scores, with every row containing the scores one person gave to all films.

This matrix then will get decomposed into three matrices:

  • (mathbf{Sigma}) shops the significance of the latent components governing the connection between customers and flicks.
  • (mathbf{U}) incorporates data on how customers rating on these latent components. It’s a illustration (embedding) of customers by the scores they gave to the flicks.
  • (mathbf{V}) shops how films rating on these identical latent components. It’s a illustration (embedding) of films by how they obtained rated by stated customers.

As quickly as we now have a illustration of films in addition to customers in the identical latent area, we are able to decide their mutual match by a easy dot product (mathbf{m^ t}mathbf{u}). Assuming the person and film vectors have been normalized to size 1, that is equal to calculating the cosine similarity

[cos(theta) = frac{mathbf{x^ t}mathbf{y}}{mathbfspacemathbf}]

What does all this need to do with embeddings?

Effectively, the identical general ideas apply once we work with person resp. film embeddings, as an alternative of vectors obtained from matrix factorization. We’ll have one layer_embedding for customers, one layer_embedding for films, and a layer_lambda that calculates the dot product.

Right here’s a minimal customized mannequin that does precisely this:

simple_dot <- perform(embedding_dim,
                       n_users,
                       n_movies,
                       identify = "simple_dot") {
  
  keras_model_custom(identify = identify, perform(self) {
    self$user_embedding <-
      layer_embedding(
        input_dim = n_users + 1,
        output_dim = embedding_dim,
        embeddings_initializer = initializer_random_uniform(minval = 0, maxval = 0.05),
        identify = "user_embedding"
      )
    self$movie_embedding <-
      layer_embedding(
        input_dim = n_movies + 1,
        output_dim = embedding_dim,
        embeddings_initializer = initializer_random_uniform(minval = 0, maxval = 0.05),
        identify = "movie_embedding"
      )
    self$dot <-
      layer_lambda(
        f = perform(x) {
          k_batch_dot(x[[1]], x[[2]], axes = 2)
        }
      )
    
    perform(x, masks = NULL) {
      customers <- x[, 1]
      films <- x[, 2]
      user_embedding <- self$user_embedding(customers)
      movie_embedding <- self$movie_embedding(films)
      self$dot(checklist(user_embedding, movie_embedding))
    }
  })
}

We’re nonetheless lacking the information although! Let’s load it.
Moreover the scores themselves, we’ll additionally get the titles from films.csv.

data_dir <- "ml-latest-small"
films <- read_csv(file.path(data_dir, "films.csv"))
scores <- read_csv(file.path(data_dir, "scores.csv"))

Whereas person ids don’t have any gaps on this pattern, that’s completely different for film ids. We due to this fact convert them to consecutive numbers, so we are able to later specify an enough measurement for the lookup matrix.

dense_movies <- scores %>% choose(movieId) %>% distinct() %>% rowid_to_column()
scores <- scores %>% inner_join(dense_movies) %>% rename(movieIdDense = rowid)
scores <- scores %>% inner_join(films) %>% choose(userId, movieIdDense, ranking, title, genres)

Let’s take a be aware, then, of what number of customers resp. films we now have.

n_movies <- scores %>% choose(movieIdDense) %>% distinct() %>% nrow()
n_users <- scores %>% choose(userId) %>% distinct() %>% nrow()

We’ll break up off 20% of the information for validation.
After coaching, most likely all customers can have been seen by the community, whereas very possible, not all films can have occurred within the coaching pattern.

train_indices <- pattern(1:nrow(scores), 0.8 * nrow(scores))
train_ratings <- scores[train_indices,]
valid_ratings <- scores[-train_indices,]

x_train <- train_ratings %>% choose(c(userId, movieIdDense)) %>% as.matrix()
y_train <- train_ratings %>% choose(ranking) %>% as.matrix()
x_valid <- valid_ratings %>% choose(c(userId, movieIdDense)) %>% as.matrix()
y_valid <- valid_ratings %>% choose(ranking) %>% as.matrix()

Coaching a easy dot product mannequin

We’re prepared to begin the coaching course of. Be at liberty to experiment with completely different embedding dimensionalities.

embedding_dim <- 64

mannequin <- simple_dot(embedding_dim, n_users, n_movies)

mannequin %>% compile(
  loss = "mse",
  optimizer = "adam"
)

historical past <- mannequin %>% match(
  x_train,
  y_train,
  epochs = 10,
  batch_size = 32,
  validation_data = checklist(x_valid, y_valid),
  callbacks = checklist(callback_early_stopping(persistence = 2))
)

How properly does this work? Ultimate RMSE (the sq. root of the MSE loss we had been utilizing) on the validation set is round 1.08 , whereas widespread benchmarks (e.g., of the LibRec recommender system) lie round 0.91. Additionally, we’re overfitting early. It appears to be like like we want a barely extra refined system.

Training curve for simple dot product model

Accounting for person and film biases

An issue with our methodology is that we attribute the ranking as a complete to user-movie interplay.
Nevertheless, some customers are intrinsically extra vital, whereas others are typically extra lenient. Analogously, movies differ by common ranking.
We hope to get higher predictions when factoring in these biases.

Conceptually, we then calculate a prediction like this:

[pred = avg + bias_m + bias_u + mathbf{m^ t}mathbf{u}]

The corresponding Keras mannequin will get simply barely extra advanced. Along with the person and film embeddings we’ve already been working with, the beneath mannequin embeds the common person and the common film in 1-d area. We then add each biases to the dot product encoding user-movie interplay.
A sigmoid activation normalizes to a worth between 0 and 1, which then will get mapped again to the unique area.

Observe how on this mannequin, we additionally use dropout on the person and film embeddings (once more, the most effective dropout fee is open to experimentation).

max_rating <- scores %>% summarise(max_rating = max(ranking)) %>% pull()
min_rating <- scores %>% summarise(min_rating = min(ranking)) %>% pull()

dot_with_bias <- perform(embedding_dim,
                          n_users,
                          n_movies,
                          max_rating,
                          min_rating,
                          identify = "dot_with_bias"
                          ) {
  keras_model_custom(identify = identify, perform(self) {
    
    self$user_embedding <-
      layer_embedding(input_dim = n_users + 1,
                      output_dim = embedding_dim,
                      identify = "user_embedding")
    self$movie_embedding <-
      layer_embedding(input_dim = n_movies + 1,
                      output_dim = embedding_dim,
                      identify = "movie_embedding")
    self$user_bias <-
      layer_embedding(input_dim = n_users + 1,
                      output_dim = 1,
                      identify = "user_bias")
    self$movie_bias <-
      layer_embedding(input_dim = n_movies + 1,
                      output_dim = 1,
                      identify = "movie_bias")
    self$user_dropout <- layer_dropout(fee = 0.3)
    self$movie_dropout <- layer_dropout(fee = 0.6)
    self$dot <-
      layer_lambda(
        f = perform(x)
          k_batch_dot(x[[1]], x[[2]], axes = 2),
        identify = "dot"
      )
    self$dot_bias <-
      layer_lambda(
        f = perform(x)
          k_sigmoid(x[[1]] + x[[2]] + x[[3]]),
        identify = "dot_bias"
      )
    self$pred <- layer_lambda(
      f = perform(x)
        x * (self$max_rating - self$min_rating) + self$min_rating,
      identify = "pred"
    )
    self$max_rating <- max_rating
    self$min_rating <- min_rating
    
    perform(x, masks = NULL) {
      
      customers <- x[, 1]
      films <- x[, 2]
      user_embedding <-
        self$user_embedding(customers) %>% self$user_dropout()
      movie_embedding <-
        self$movie_embedding(films) %>% self$movie_dropout()
      dot <- self$dot(checklist(user_embedding, movie_embedding))
      dot_bias <-
        self$dot_bias(checklist(dot, self$user_bias(customers), self$movie_bias(films)))
      self$pred(dot_bias)
    }
  })
}

How properly does this mannequin carry out?

mannequin <- dot_with_bias(embedding_dim,
                       n_users,
                       n_movies,
                       max_rating,
                       min_rating)

mannequin %>% compile(
  loss = "mse",
  optimizer = "adam"
)

historical past <- mannequin %>% match(
  x_train,
  y_train,
  epochs = 10,
  batch_size = 32,
  validation_data = checklist(x_valid, y_valid),
  callbacks = checklist(callback_early_stopping(persistence = 2))
)

Not solely does it overfit later, it really reaches a method higher RMSE of 0.88 on the validation set!

Training curve for dot product model with biases

Spending a while on hyperparameter optimization might very properly result in even higher outcomes.
As this submit focuses on the conceptual facet although, we need to see what else we are able to do with these embeddings.

Embeddings: a more in-depth look

We will simply extract the embedding matrices from the respective layers. Let’s do that for films now.

movie_embeddings <- (mannequin %>% get_layer("movie_embedding") %>% get_weights())[[1]]

How are they distributed? Right here’s a heatmap of the primary 20 films. (Observe how we increment the row indices by 1, as a result of the very first row within the embedding matrix belongs to a film id 0 which doesn’t exist in our dataset.)
We see that the embeddings look quite uniformly distributed between -0.5 and 0.5.

levelplot(
  t(movie_embeddings[2:21, 1:64]),
  xlab = "",
  ylab = "",
  scale = (checklist(draw = FALSE)))
Embeddings for first 20 movies

Naturally, we could be all in favour of dimensionality discount, and see how particular films rating on the dominant components.
A potential option to obtain that is PCA:

movie_pca <- movie_embeddings %>% prcomp(heart = FALSE)
parts <- movie_pca$x %>% as.knowledge.body() %>% rowid_to_column()

plot(movie_pca)
PCA: Variance explained by component

Let’s simply take a look at the primary principal part as the second already explains a lot much less variance.

Listed below are the ten films (out of all that had been rated at the least 20 occasions) that scored lowest on the primary issue:

ratings_with_pc12 <-
  scores %>% inner_join(parts %>% choose(rowid, PC1, PC2),
                         by = c("movieIdDense" = "rowid"))

ratings_grouped <-
  ratings_with_pc12 %>%
  group_by(title) %>%
  summarize(
    PC1 = max(PC1),
    PC2 = max(PC2),
    ranking = imply(ranking),
    genres = max(genres),
    num_ratings = n()
  )

ratings_grouped %>% filter(num_ratings > 20) %>% prepare(PC1) %>% print(n = 10)
# A tibble: 1,247 x 6
   title                                   PC1      PC2 ranking genres                   num_ratings
   <chr>                                 <dbl>    <dbl>  <dbl> <chr>                          <int>
 1 Starman (1984)                       -1.15  -0.400     3.45 Journey|Drama|Romance…          22
 2 Bulworth (1998)                      -0.820  0.218     3.29 Comedy|Drama|Romance              31
 3 Cable Man, The (1996)                -0.801 -0.00333   2.55 Comedy|Thriller                   59
 4 Species (1995)                       -0.772 -0.126     2.81 Horror|Sci-Fi                     55
 5 Save the Final Dance (2001)           -0.765  0.0302    3.36 Drama|Romance                     21
 6 Spanish Prisoner, The (1997)         -0.760  0.435     3.91 Crime|Drama|Thriller|Thr…          23
 7 Sgt. Bilko (1996)                    -0.757  0.249     2.76 Comedy                            29
 8 Bare Gun 2 1/2: The Odor of Worry,… -0.749  0.140     3.44 Comedy                            27
 9 Swordfish (2001)                     -0.694  0.328     2.92 Motion|Crime|Drama                33
10 Addams Household Values (1993)          -0.693  0.251     3.15 Youngsters|Comedy|Fantasy           73
# ... with 1,237 extra rows

And right here, inversely, are those who scored highest:

ratings_grouped %>% filter(num_ratings > 20) %>% prepare(desc(PC1)) %>% print(n = 10)
 A tibble: 1,247 x 6
   title                                PC1        PC2 ranking genres                    num_ratings
   <chr>                              <dbl>      <dbl>  <dbl> <chr>                           <int>
 1 Graduate, The (1967)                1.41  0.0432      4.12 Comedy|Drama|Romance               89
 2 Vertigo (1958)                      1.38 -0.0000246   4.22 Drama|Thriller|Romance|Th…          69
 3 Breakfast at Tiffany's (1961)       1.28  0.278       3.59 Drama|Romance                      44
 4 Treasure of the Sierra Madre, The…  1.28 -0.496       4.3  Motion|Journey|Drama|W…          30
 5 Boot, Das (Boat, The) (1981)        1.26  0.238       4.17 Motion|Drama|Struggle                   51
 6 Flintstones, The (1994)             1.18  0.762       2.21 Youngsters|Comedy|Fantasy            39
 7 Rock, The (1996)                    1.17 -0.269       3.74 Motion|Journey|Thriller         135
 8 Within the Warmth of the Evening (1967)     1.15 -0.110       3.91 Drama|Thriller                      22
 9 Quiz Present (1994)                    1.14 -0.166       3.75 Drama                              90
10 Striptease (1996)                   1.14 -0.681       2.46 Comedy|Crime                       39
# ... with 1,237 extra rows

We’ll go away it to the educated reader to call these components, and proceed to our second matter: How does an embedding layer do what it does?

Do-it-yourself embeddings

You will have heard individuals say all an embedding layer did was only a lookup. Think about you had a dataset that, along with steady variables like temperature or barometric strain, contained a categorical column characterization consisting of tags like “foggy” or “cloudy.” Say characterization had 7 potential values, encoded as an element with ranges 1-7.

Have been we going to feed this variable to a non-embedding layer, layer_dense say, we’d need to take care that these numbers don’t get taken for integers, thus falsely implying an interval (or at the least ordered) scale. However once we use an embedding as the primary layer in a Keras mannequin, we feed in integers on a regular basis! For instance, in textual content classification, a sentence would possibly get encoded as a vector padded with zeroes, like this:

2  77   4   5 122   55  1  3   0   0  

The factor that makes this work is that the embedding layer really does carry out a lookup. Beneath, you’ll discover a quite simple customized layer that does basically the identical factor as Keras’ layer_embedding:

  • It has a weight matrix self$embeddings that maps from an enter area (films, say) to the output area of latent components (embeddings).
  • After we name the layer, as in

x <- k_gather(self$embeddings, x)

it appears to be like up the passed-in row quantity within the weight matrix, thus retrieving an merchandise’s distributed illustration from the matrix.

SimpleEmbedding <- R6::R6Class(
  "SimpleEmbedding",
  
  inherit = KerasLayer,
  
  public = checklist(
    output_dim = NULL,
    emb_input_dim = NULL,
    embeddings = NULL,
    
    initialize = perform(emb_input_dim, output_dim) {
      self$emb_input_dim <- emb_input_dim
      self$output_dim <- output_dim
    },
    
    construct = perform(input_shape) {
      self$embeddings <- self$add_weight(
        identify = 'embeddings',
        form = checklist(self$emb_input_dim, self$output_dim),
        initializer = initializer_random_uniform(),
        trainable = TRUE
      )
    },
    
    name = perform(x, masks = NULL) {
      x <- k_cast(x, "int32")
      k_gather(self$embeddings, x)
    },
    
    compute_output_shape = perform(input_shape) {
      checklist(self$output_dim)
    }
  )
)

As common with customized layers, we nonetheless want a wrapper that takes care of instantiation.

layer_simple_embedding <-
  perform(object,
           emb_input_dim,
           output_dim,
           identify = NULL,
           trainable = TRUE) {
    create_layer(
      SimpleEmbedding,
      object,
      checklist(
        emb_input_dim = as.integer(emb_input_dim),
        output_dim = as.integer(output_dim),
        identify = identify,
        trainable = trainable
      )
    )
  }

Does this work? Let’s take a look at it on the scores prediction job! We’ll simply substitute the customized layer within the easy dot product mannequin we began out with, and test if we get out the same RMSE.

Placing the customized embedding layer to check

Right here’s the easy dot product mannequin once more, this time utilizing our customized embedding layer.

simple_dot2 <- perform(embedding_dim,
                       n_users,
                       n_movies,
                       identify = "simple_dot2") {
  
  keras_model_custom(identify = identify, perform(self) {
    self$embedding_dim <- embedding_dim
    
    self$user_embedding <-
      layer_simple_embedding(
        emb_input_dim = checklist(n_users + 1),
        output_dim = embedding_dim,
        identify = "user_embedding"
      )
    self$movie_embedding <-
      layer_simple_embedding(
        emb_input_dim = checklist(n_movies + 1),
        output_dim = embedding_dim,
        identify = "movie_embedding"
      )
    self$dot <-
      layer_lambda(
        output_shape = self$embedding_dim,
        f = perform(x) {
          k_batch_dot(x[[1]], x[[2]], axes = 2)
        }
      )
    
    perform(x, masks = NULL) {
      customers <- x[, 1]
      films <- x[, 2]
      user_embedding <- self$user_embedding(customers)
      movie_embedding <- self$movie_embedding(films)
      self$dot(checklist(user_embedding, movie_embedding))
    }
  })
}

mannequin <- simple_dot2(embedding_dim, n_users, n_movies)

mannequin %>% compile(
  loss = "mse",
  optimizer = "adam"
)

historical past <- mannequin %>% match(
  x_train,
  y_train,
  epochs = 10,
  batch_size = 32,
  validation_data = checklist(x_valid, y_valid),
  callbacks = checklist(callback_early_stopping(persistence = 2))
)

We find yourself with a RMSE of 1.13 on the validation set, which isn’t removed from the 1.08 we obtained when utilizing layer_embedding. A minimum of, this could inform us that we efficiently reproduced the method.

Conclusion

Our targets on this submit had been twofold: Shed some gentle on how an embedding layer could be carried out, and present how embeddings calculated by a neural community can be utilized as an alternative to part matrices obtained from matrix decomposition. In fact, this isn’t the one factor that’s fascinating about embeddings!

For instance, a really sensible query is how a lot precise predictions could be improved through the use of embeddings as an alternative of one-hot vectors; one other is how realized embeddings would possibly differ relying on what job they had been skilled on.
Final not least – how do latent components realized through embeddings differ from these realized by an autoencoder?

In that spirit, there isn’t any lack of subjects for exploration and poking round …

Ahmed, N. Okay., R. Rossi, J. Boaz Lee, T. L. Willke, R. Zhou, X. Kong, and H. Eldardiry. 2018. “Studying Function-Primarily based Graph Embeddings.” ArXiv e-Prints, February. https://arxiv.org/abs/1802.02896.
Alon, Uri, Meital Zilberstein, Omer Levy, and Eran Yahav. 2018. “Code2vec: Studying Distributed Representations of Code.” CoRR abs/1803.09473. http://arxiv.org/abs/1803.09473.

Frome, Andrea, Gregory S. Corrado, Jonathon Shlens, Samy Bengio, Jeffrey Dean, Marc’Aurelio Ranzato, and Tomas Mikolov. 2013. “DeViSE: A Deep Visible-Semantic Embedding Mannequin.” In NIPS, 2121–29.

Hallac, D., S. Bhooshan, M. Chen, Okay. Abida, R. Sosic, and J. Leskovec. 2018. “Drive2Vec: Multiscale State-Area Embedding of Vehicular Sensor Information.” ArXiv e-Prints, June. https://arxiv.org/abs/1806.04795.
Jean, Neal, Sherrie Wang, Anshul Samar, George Azzari, David B. Lobell, and Stefano Ermon. 2018. “Tile2Vec: Unsupervised Illustration Studying for Spatially Distributed Information.” CoRR abs/1805.02855. http://arxiv.org/abs/1805.02855.
Krstovski, Okay., and D. M. Blei. 2018. “Equation Embeddings.” ArXiv e-Prints, March. https://arxiv.org/abs/1803.09123.

Rumelhart, David E., James L. McClelland, and CORPORATE PDP Analysis Group, eds. 1986. Parallel Distributed Processing: Explorations within the Microstructure of Cognition, Vol. 2: Psychological and Organic Fashions. Cambridge, MA, USA: MIT Press.

Zohra Smaili, F., X. Gao, and R. Hoehndorf. 2018. “Onto2Vec: Joint Vector-Primarily based Illustration of Organic Entities and Their Ontology-Primarily based Annotations.” ArXiv e-Prints, January. https://arxiv.org/abs/1802.00864.

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