Basically, this question is some sort of research question. I post it here to gain more hints from you. Suppose you have a set S of n d-dimension vectors. Given a d-dimension query vector, I want to find two vectors from S whose sum is close to the query vector, in term of cosine distance.
Right now, my solution is enumerating all possible combinations of S, results to n^2 possible sum-vectors. Then hash these n^2 vectors into b buckets using Local sensitive hashing. When given the query vector, we hash the query into one of the b buckets and check all the possible sum-vectors in that bucket, to see which one is closest. This will need O(b) space and O(n^2/b) searching time. This is just an space time trade off.
Is there any other good solutions ? Any hints are welcome ! I don't need the exact closest results, approximation results are more welcome!
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I have about 500 vectors,each vector is a 1500-dimension vector,
and almost every vector is very sparse-- I mean only about 30-70 dimension of the vector is not 0。
Now, the problom is that here is a given vetor,also 1500 dimension,and I need to compare it to the 500 vectors to find which of the 500 is the nearest one.(In euclidean distance).
There is no doubt that brute-force method is a solution , but I need to calculate the distance for 500 times ,which takes a long time.
Yesterday I read an article "Object retrieval with large vocabularies and fast spatial matching", it says using inverted index will help,its says:
but after my test, it made almost no sense, imagine a 1500-vector in which 50 of the dimension are not zero, when it comes to another one, they may always have the same dimension that are not zero. In other words, this algorithm can only rule out a little vectors, I still need to compare with many vectors left.
Thank you for your nice that you have read to here, my question is that:
1.will this algorithm make sense?
2.is there any other way to do what I want to do? such as flann or Kd-TREE?
but I want the exact accurate nearest neighbor, a approxiate one is not enough
This kind of index is called inverted lists, and is commonly used for text.
For example, Apache Lucene uses this kind of indexing for text similarity search.
Essentially, you use a columnar layout, and you only store the non-zero values. For on-disk efficiency, various compression techniques can be employed.
You can then compute many similarities using set operations on these lists.
k-d-trees cannot be used here. They will be extremely inefficient if you have many duplicate (zero) values.
I don't know your context but if you don't care of having a long preprocess step and you have to make this check often and fast, you can build a neighborhood graph and sorting neighbors by distances.
To efficiently build this graph you can perform a taxicab distance or a square distance to sort the points by distances (This will avoid heavy calculations).
Then if you want the nearest neighbor you just have to pick the first neighbor :p.
"LSH has some huge advantages. First, it is simple. You just compute the hash for all
points in your database, then make a hash table from them. To query, just compute the hash of
the query point, then retrieve all points in the same bin from the hash table."
Referring to the answer on another question I am looking for clarification of the process of LSH analysis.
Suppose I have sparse feature vectors (binary, mostly 0) and would like to use cosine distance as the measure with a threshold alpha, which might vary.
My first step is to compute the hash for each of the vectors. Does distance measure matter? (I suppose yes). Does threshold matters? (I suppose no). How can I find the appropriate hash-function?
If programming, I would have function like:
bytes[] getHash(Vector featureVec)
Then I would put results in the Map(long vectorId, bytes[] hashcode) <-vectorHashMap
Then I make hash table from hashes (putting hashs into bins). I suppose at least here should the threshold matter. How can I do that?
If programming, it would be like:
Map,Map createHashTable(Map vectorHashMap, long threshold)
which returns two maps: Map of (hashCode, bucketId) and Map of (bucketId, ListOfVectorIds).
Then i could easily retrieve the neigbors having vectorId as input and a list of vectorIds as output.
The hash has nothing to do with the distance measure. You get the each bit of the hash by dotting the vector with a randomly-chosen vector. The bit says what side of the random vector (a hyperplane really) the hashed vector is on. The bits together are a hash.
Yes then you index your vectors by hash value for easy retrieval. You don't need a 'bucket ID' -- your hash is your bucket.
The only catch here is that it is not true that all the nearest vectors are in the bucket that it hashed too. They merely tend to be near. If it mattered, you might have to search 'similar' buckets -- ones that differ in just a few bits -- to consider more candidates and do a better job of finding truly closest neighbors.
Currently I'm studying how to find a nearest neighbor using Locality-sensitive hashing. However while I'm reading papers and searching the web I found two algorithms for doing this:
1- Use L number of hash tables with L number of random LSH functions, thus increasing the chance that two documents that are similar to get the same signature. For example if two documents are 80% similar, then there's an 80% chance that they will get the same signature from one LSH function. However if we use multiple LSH functions, then there's a higher chance to get the same signature for the documents from one of the LSH functions. This method is explained in wikipedia and I hope my understanding is correct:
http://en.wikipedia.org/wiki/Locality-sensitive_hashing#LSH_algorithm_for_nearest_neighbor_search
2- The other algorithm uses a method from a paper (section 5) called: Similarity Estimation Techniques from Rounding Algorithms by Moses S. Charikar. It's based on using one LSH function to generate the signature and then apply P permutations on it and then sort the list. Actually I don't understand the method very well and I hope if someone could clarify it.
My main question is: why would anyone use the second method rather than the first method? As I find it's easier and faster.
I really hope someone can help!!!
EDIT:
Actually I'm not sure if #Raff.Edward were mixing between the "first" and the "second". Because only the second method uses a radius and the first just uses a new hash family g composed of the hash family F. Please check the wikipedia link. They just used many g functions to generate different signatures and then for each g function it has a corresponding hash table. In order to find the nearest neighbor of a point you just let the point go through the g functions and check the corresponding hash tables for collisions. Thus how I understood it as more function ... more chance for collisions.
I didn't find any mentioning about radius for the first method.
For the second method they generate only one signature for each feature vector and then apply P permutations on them. Now we have P lists of permutations where each contains n signatures. Now they then sort each list from P. After that given a query point q, they generate the signature for it and then apply the P permutations on it and then use binary search on each permuted and sorted P list to find the most similar signature to the query q. I concluded this after reading many papers about it, but I still don't understand why would anyone use such a method because it doesn't seem fast in finding the hamming distance!!!!
For me I would simply do the following to find the nearest neighbor for a query point q. Given a list of signatures N, I would generate the signature for the query point q and then scan the list N and compute the hamming distance between each element in N and the signature of q. Thus I would end up with the nearest neighbor for q. And it takes O(N)!!!
Your understanding of the first one is a little off. The probability of a collision occurring is not proportional to the similarity, but whether or not it is less than the pre-defined radius. The goal is that anything within the radius will have a high chance of colliding, and anything outside the radius * (1+eps) will have a low chance of colliding (and the area in-between is a little murky).
The first algorithm is actually fairly difficult to implement well, but can get good results. In particular, the first algorithm is for the L1 and L2 (and technically a few more) metrics.
The second algorithm is very simple to implement, though a naive implementation may use up too much memory to be useful depending on your problem size. In this case, the probability of collision is proportional to the similarity of the inputs. However, it only works for the Cosine Similarity (or distance metrics based on a transform of the similarity.)
So which one you would use is based primarily on which distance metric you are using for Nearest Neighbor (or whatever other application).
The second one is actually much easier to understand and implement than the first one, the paper is just very wordy.
The short version: Take a random vector V and give each index a independent random unit normal value. Create as many vectors as you want the signature length to be. The signature is the signs of each index when you do a Matrix Vector product. Now the hamming distance between any two signatures is related to the cosine similarity between the respective data points.
Because you can encode the signature into an int array and use an XOR with a bit count instruction to get the hamming distance very quickly, you can get approximate cosine similarity scores very quickly.
LSH algorithms doesn't have a lot of standardization, and the two papers (and others) use different definitions, so its all a bit confusing at times. I only recently implemented both of these algorithms in JSAT, and am still working on fully understanding them both.
EDIT: Replying to your edit. The wikipedia article is not great for LSH. If you read the original paper, the first method you are talking about only works for a fixed radius. The hash functions are then created based on that radius, and concatenated to increase the probability of getting near by points in a collision. They then construct a system for doing k-NN on-top of this by determine the maximum value of k they wan, and then finding the largest reasonable distance they would find the k'th nearest neighbor in. In this way, a radius search will very likely return the set of k-NNs. To speed this up, they also create a few extra small radius since the density is often not uniform, and the smaller radius you use, the faster the results.
The wikipedia section you linked is taken from the paper description for the "Stable Distribution" section, which presents the hash function for a search of radius r=1.
For the second paper, the "sorting" you describe is not part of the hashing, but part of one-scheme for searching the hamming space more quickly. I as I mentioned, I recently implemented this, and you can see a quick benchmark I did using a brute force search is still much faster than the naive method of NN. Again, you would also pick this method if you need the cosine similarity over the L2 or L1 distance. You will find many other papers proposing different schemes for searching the hamming space created by the signatures.
If you need help convincing yourself fit can be faster even if you were still doing brute force - just look at it this way: Lets say that the average sparse document has 40 common words with another document (a very conservative number in my experience). You have n documents to compare against. Brute force cosine similarity would then involve about 40*n floating point multiplications (and some extra work). If you have a 1024 bit signature, thats only 32 integers. That means we could do a brute force LSH search in 32*n integer operations, which are considerably faster then floating point operations.
There are also other factors at play here as well. For a sparse data set we have to keep both the doubles and integer indices to represent the non zero indexes, so the sparse dot product is doing a lot of additional integer operations to see which indices they have in common. LSH also allows us to save memory, because we don't need to store all of these integers and doubles for each vector, instead we can just keep its hash around - which is only a few bytes.
Reduced memory use can help us better exploit the CPU cache.
Your O(n) is the naive way I have used in my blog post. And it is fast. However, if you sort the bits before hand, you can do the binary search in O(log(n)). Even if you have L of these lists, L << n, and so it should be faster. The only issue is it gets you approximate hamming NN which are already approximating the cosine similarity, so the results can become a bit worse. It depends on what you need.
I have about 100M numeric vectors (Minhash fingerprints), each vector contains 100 integer numbers between 0 and 65536, and I'm trying to do a fast similarity search against this database of fingerprints using Jaccard similarity, i.e. given a query vector (e.g. [1,0,30, 9, 42, ...]) find the ratio of intersection/union of this query set against the database of 100M sets.
The requirement is to return k "nearest neighbors" of the query vector in <1 sec (not including indexing/File IO time) on a laptop. So obviously some kind of indexing is required, and the question is what would be the most efficient way to approach this.
notes:
I thought of using SimHash but in this case actually need to know the size of intersection of the sets to identify containment rather than pure similarity/resemblance, but Simhash would lose that information.
I've tried using a simple locality sensitive hashing technique as described in ch3 of Jeffrey Ullman's book by dividing each vector into 20 "bands" or snippets of length 5, converting these snippets into strings (e.g. [1, 2, 45, 2, 3] - > "124523") and using these strings as keys in a hash table, where each key contains "candidate neighbors". But the problem is that it creates too many candidates for some of these snippets and changing number of bands doesn't help.
I might be a bit late, but I would suggest IVFADC indexing by Jegou et al.: Product Quantization for Nearest Neighbor Search
It works for L2 Distance/dot product similarity measures and is a bit complex, but it's particularly efficient in terms of both time and memory.
It is also implemented in the FAISS library for similarity search, so you could also take a look at that.
One way to go about this is the following:
(1) Arrange the vectors into a tree (a radix tree).
(2) Query the tree with a fuzzy criteria, in other words, a match is if the difference in values at each node of the tree is within a threshold
(3) From (2) generate a subtree that contains all the matching vectors
(4) Now, repeat process (2) on the sub tree with a smaller threshold
Continue until the subtree has K items. If K has too few items, then take the previous tree and calculate the Jacard distance on each member of the subtree and sort to eliminate the worst matches until you have only K items left.
answering my own question after 6 years, there is a benchmark for approximate nearest neighbor search with many algorithms to solve this problem: https://github.com/erikbern/ann-benchmarks, the current winner is "Hierarchical Navigable Small World graphs": https://github.com/nmslib/hnswlib
You can use off-the-shelf similarity search services such as AWS-ES or Pinecone.io.
I've got a MongoDB with about 1 million documents in it. These documents all have a string that represents a 256 bit bin of 1s and 0s, like:
0110101010101010110101010101
Ideally, I'd like to query for near binary matches. This means, if the two documents have the following numbers. Yes, this is Hamming Distance.
This is NOT currently supported in Mongo. So, I'm forced to do it in the application layer.
So, given this, I am trying to find a way to avoid having to do individual Hamming distance comparisons between the documents. that makes the time to do this basically impossible.
I have a LOT of RAM. And, in ruby, there seems to be a great gem (algorithms) that can create a number of trees, none of which I can seem to make work (yet) that would reduce the number of queries I'd need to make.
Ideally, I'd like to make 1 million queries, find the near duplicate strings, and be able to update them to reflect that.
Anyone's thoughts would be appreciated.
I ended up doing a retrieval of all the documents into memory.. (subset with the id and the string).
Then, I used a BK Tree to compare the strings.
The Hamming distance defines a metric space, so you could use the O(n log n) algorithm to find the closest pair of points, which is of the typical divide-and-conquer nature.
You can then apply this repeatedly until you have "enough" pairs.
Edit: I see now that Wikipedia doesn't actually give the algorithm, so here is one description.
Edit 2: The algorithm can be modified to give up if there are no pairs at distance less than n. For the case of the Hamming distance: simply count the level of recursion you are in. If you haven't found something at level n in any branch, then give up (in other words, never enter n + 1). If you are using a metric where splitting on one dimension doesn't always yield a distance of 1, you need to adjust the level of recursion where you give up.
As far as I could understand, you have an input string X and you want to query the database for a document containing string field b such that Hamming distance between X and document.b is less than some small number d.
You can do this in linear time, just by scanning all of your N=1M documents and calculating the distance (which takes small fixed time per document). Since you only want documents with distance smaller than d, you can give up comparison after d unmatched characters; you only need to compare all 256 characters if most of them match.
You can try to scan fewer than N documents, that is, to get better than linear time.
Let ones(s) be the number of 1s in string s. For each document, store ones(document.b) as a new indexed field ones_count. Then you can only query documents where number of ones is close enough to ones(X), specifically, ones(X) - d <= document.ones_count <= ones(X) + d. The Mongo index should kick in here.
If you want to find all close enough pairs in the set, see #Philippe's answer.
This sounds like an algorithmic problem of some sort. You could try comparing those with a similar number of 1 or 0 bits first, then work down through the list from there. Those that are identical will, of course, come out on top. I don't think having tons of RAM will help here.
You could also try and work with smaller chunks. Instead of dealing with 256 bit sequences, could you treat that as 32 8-bit sequences? 16 16-bit sequences? At that point you can compute differences in a lookup table and use that as a sort of index.
Depending on how "different" you care to match on, you could just permute changes on the source binary value and do a keyed search to find the others that match.