Smooth algebra

In machine learning, feature hashing, also known as the hashing trick (by analogy to the kernel trick), is a fast and space-efficient way of vectorizing features, i.e. turning arbitrary features into indices in a vector or matrix. It works by applying a hash function to the features and using their hash values as indices directly, rather than looking the indices up in an associative array.

Motivating example

In a typical document classification task, the input to the machine learning algorithm (both during learning and classification) is free text. From this, a bag of words (BOW) representation is constructed: the individual tokens are extracted and counted, and each distinct token in the training set defines a feature (dependent variable) of each of the documents in both the training and test sets.

Machine learning algorithms, however, are typically defined in terms of numerical vectors. Therefore, the bags of words for a set of documents is regarded as a term-document matrix where each row is a single document, and each column is a single feature/word; the entry Buying, selling and renting HDB and personal residential properties in Singapore are simple and transparent transactions. Although you are not required to engage a real property salesperson (generally often known as a "public listed property developers In singapore agent") to complete these property transactions, chances are you'll think about partaking one if you are not accustomed to the processes concerned.

Professional agents are readily available once you need to discover an condominium for hire in singapore In some cases, landlords will take into account you more favourably in case your agent comes to them than for those who tried to method them by yourself. You need to be careful, nevertheless, as you resolve in your agent. Ensure that the agent you are contemplating working with is registered with the IEA – Institute of Estate Brokers. Whereas it might sound a hassle to you, will probably be worth it in the end. The IEA works by an ordinary algorithm and regulations, so you'll protect yourself in opposition to probably going with a rogue agent who prices you more than they should for his or her service in finding you an residence for lease in singapore.

There isn't any deal too small. Property agents who are keen to find time for any deal even if the commission is small are the ones you want on your aspect. Additionally they present humbleness and might relate with the typical Singaporean higher. Relentlessly pursuing any deal, calling prospects even without being prompted. Even if they get rejected a hundred times, they still come again for more. These are the property brokers who will find consumers what they need eventually, and who would be the most successful in what they do. 4. Honesty and Integrity

This feature is suitable for you who need to get the tax deductions out of your PIC scheme to your property agency firm. It's endorsed that you visit the correct site for filling this tax return software. This utility must be submitted at the very least yearly to report your whole tax and tax return that you're going to receive in the current accounting 12 months. There may be an official website for this tax filling procedure. Filling this tax return software shouldn't be a tough thing to do for all business homeowners in Singapore.

A wholly owned subsidiary of SLP Worldwide, SLP Realty houses 900 associates to service SLP's fast rising portfolio of residential tasks. Real estate is a human-centric trade. Apart from offering comprehensive coaching applications for our associates, SLP Realty puts equal emphasis on creating human capabilities and creating sturdy teamwork throughout all ranges of our organisational hierarchy. Worldwide Presence At SLP International, our staff of execs is pushed to make sure our shoppers meet their enterprise and investment targets. Under is an inventory of some notable shoppers from completely different industries and markets, who've entrusted their real estate must the expertise of SLP Worldwide.

If you're looking for a real estate or Singapore property agent online, you merely need to belief your instinct. It is because you don't know which agent is sweet and which agent will not be. Carry out research on a number of brokers by looking out the internet. As soon as if you find yourself certain that a selected agent is dependable and trustworthy, you'll be able to choose to utilize his partnerise find you a house in Singapore. More often than not, a property agent is considered to be good if she or he places the contact data on his web site. This is able to imply that the agent does not thoughts you calling them and asking them any questions regarding properties in Singapore. After chatting with them you too can see them of their office after taking an appointment.

Another method by way of which you could find out whether the agent is sweet is by checking the feedback, of the shoppers, on the website. There are various individuals would publish their comments on the web site of the Singapore property agent. You can take a look at these feedback and the see whether it will be clever to hire that specific Singapore property agent. You may even get in contact with the developer immediately. Many Singapore property brokers know the developers and you may confirm the goodwill of the agent by asking the developer. in such a matrix captures the frequency (or weight) of the Template:Mvar'th term of the vocabulary in document Template:Mvar. (An alternative convention swaps the rows and columns of the term-document matrix, but this difference is immaterial.) Typically, these vectors are extremely sparse.

The common approach is to construct, at learning time or prior to that, a dictionary representation of the vocabulary of the training set, and use that to map words to indices. Hash tables and tries are common candidates for dictionary implementation. E.g., the texts

John likes to watch movies. Mary likes movies too.
John also likes to watch football games.
An apple a day keeps the doctor away.

can be converted, using the dictionary (in Python notation):

{"John": 1, "likes": 2, "to": 3, "watch": 4, "movies": 5, "also": 6, "football": 7, "games": 8, "Mary": 9, "too": 10,
 "an": 11, "apple": 12, "a": 13, "day": 14, "keeps": 15, "the": 16, "doctor": 17, "away": 18}

to the matrix

[[1 2 1 1 2 0 0 0 1 1 0 0 0 0 0 0 0 0]
 [1 1 1 1 0 1 1 1 0 0 0 0 0 0 0 0 0 0]
 [0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1]

(Punctuation was removed, as is usual in document classification and clustering.)

The problem with this process is that such dictionaries take up a large amount of storage space, and grow in size as the training set grows. This can be seen in the above example: the third "document" has no terms in common with the other two, and therefore extends the vocabulary with its eight distinct words. Moreover, when the vocabulary is kept fixed, an adversary may try to invent new words or misspellings that are not in the stored vocabulary so as to circumvent a machine learned filter; this is why feature hashing has been tried for spam filtering at Yahoo! Research.

Note that the hashing trick isn't limited to text classification and similar tasks at the document level, but can be applied to any problem that involves large (perhaps unbounded) amounts of features.

Feature vectorization using the hashing trick

Instead of maintaining a dictionary, a feature vectorizer that uses the hashing trick can build a vector of a pre-defined length by applying a hash function Template:Mvar to the features (e.g., words) in the items under consideration, then using the hash values directly as feature indices and updating the resulting vector at those indices:

 function hashing_vectorizer(features : array of string, N : integer):
     x := new vector[N]
     for f in features:
         h := hash(f)
         x[h mod N] += 1
     return x

It has been suggested that a second, single-bit output hash function Template:Mvar be used to determine the sign of the update value, to counter the effect of hash collisions. If such a hash function is used, the algorithm becomes

 function hashing_vectorizer(features : array of string, N : integer):
     x := new vector[N]
     for f in features:
         h := hash(f)
         idx := h mod N
         if ξ(f) == 1:
             x[idx] += 1
         else:
             x[idx] -= 1
     return x

The above pseudocode actually converts each sample into a vector. An optimized version would instead only generate a stream of (Template:Mvar,Template:Mvar) pairs and let the learning and prediction algorithms consume such streams; a linear model can then be implemented as a single hash table representing the coefficient vector.

Properties

When a second hash function ξ is used to determine the sign of a feature's value, the expected mean of each column in the output array becomes zero because ξ causes some collisions to cancel out. E.g., suppose an input contains two symbolic features f₁ and f₂ that collide with each other, but not with any other features in the same input; then there are four possibilities which, if we make no assumptions about ξ, have equal probability:

ξ(f₁)	ξ(f₂)	Final value, ξ(f₁) + ξ(f₂)
-1	-1	-2
-1	1	0
1	-1	0
1	1	2

In this example, there is a 50% probability that the hash collision cancels out. Multiple hash functions can be used to further reduce the risk of collisions.

Furthermore, if φ is the transformation implemented by a hashing trick with a sign hash ξ (i.e. φ(x) is the feature vector produced for a sample x), then inner products in the hashed space are unbiased:

$\mathbb{𝔼}[⟨\phi (x),\phi (x^{\prime })⟩]=⟨x,x^{\prime }⟩$

where the expectation is taken over the hashing function φ. It can be verified that$⟨\phi (x),\phi (x^{\prime })⟩$ is a positive semi-definite kernel.

Extensions and variations

Recent work extends the hashing trick to supervised mappings from words to indices, which are explicitly learned to avoid collisions of important terms.

Applications and practical performance

Ganchev and Dredze showed that in text classification applications with random hash functions and several tens of thousands of columns in the output vectors, feature hashing need not have an adverse effect on classification performance, even without the signed hash function. Weinberger et al. applied their variant of hashing to the problem of spam filtering, formulating this as a multi-task learning problem where the input features are pairs (user, feature) so that a single parameter vector captured per-user spam filters as well as a global filter for several hundred thousand users, and found that the accuracy of the filter went up.

Implementations

Implementations of the hashing trick are present in:

• Apache Mahout
• Gensim (0.8.6 changelog)
• scikit-learn (Feature hashing)
• sofia-ml
• Vowpal Wabbit

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

See also

• Bloom filter
• Count–min sketch
• Locality-sensitive hashing
• MinHash

External links

• Hashing Representations for Machine Learning on John Langford's website
• What is the "hashing trick"? - MetaOptimize Q+A