Push fine-tuned retriever model (all-MiniLM-L6-v2 base)

2aa5e4a verified 7 months ago

37.3 kB

	---
	tags:
	- sentence-transformers
	- sentence-similarity
	- feature-extraction
	- generated_from_trainer
	- dataset_size:100000
	- loss:TripletLoss
	base_model: sentence-transformers/all-MiniLM-L6-v2
	widget:
	- source_sentence: 'Consider the set of points $S = \{(x,y) : x \text{ and } y \text{
	are non-negative integers } \leq n\}$. Find the number of squares that can be
	formed with vertices belonging to $S$ and sides parallel to the axes.'
	sentences:
	- <page_title> Waller Plan </page_title> <path> Waller_Plan > City plan </path>
	<section_title> City plan </section_title> <content> The plan also designated
	spaces for a hospital, an academy and university, churches, a courthouse and jail,
	an armory, and a penitentiary.With the surveying and grid plan completed, Waller
	and his associates drew up a plat dividing the city blocks into land lots. The
	first auction of lots was held on August 1, 1839, under a group of live oak trees
	in what was to be the city's southwestern public square; these trees have since
	been known as the "Auction Oaks". The auction raised $182,585 (equivalent to $5,018,000
	in 2022), funds used to pay for the construction of government buildings for the
	new capital city. </content>
	- <page_title> Heilbronn triangle problem </page_title> <path> Heilbronn_triangle_problem
	> Specific shapes and numbers </path> <section_title> Specific shapes and numbers
	</section_title> <content> Goldberg (1972) has investigated the optimal arrangements
	of n {\displaystyle n} points in a square, for n {\displaystyle n} up to 16. Goldberg's
	constructions for up to six points lie on the boundary of the square, and are
	placed to form an affine transformation of the vertices of a regular polygon.
	For larger values of n {\displaystyle n} , Comellas & Yebra (2002) improved Goldberg's
	bounds, and for these values the solutions include points interior to the square.
	These constructions have been proven optimal for up to seven points. </content>
	- '<page_title> 14:9 aspect ratio </page_title> <path> 14:9_aspect_ratio > Mathematics
	</path> <section_title> Mathematics </section_title> <content> The aspect ratio
	of 14:9 (1.555...) is the arithmetic mean (average) of 16:9 and 4:3 (12:9), (
	( 16 / 9 ) + ( 12 / 9 ) ) ÷ 2 = 14 / 9 {\displaystyle ((16/9)+(12/9))\div 2=14/9}
	. More practically, it is approximately the geometric mean (the precise geometric
	mean is ( 16 / 9 ) × ( 4 / 3 ) ≈ 1.5396 ≈ 13.8: 9 {\displaystyle {\sqrt {(16/9)\times
	(4/3)}}\approx 1.5396\approx 13.8:9} ), and in this sense is mathematically a
	compromise between these two aspect ratios: two equal area pictures (at 16:9 and
	4:3) will intersect in a box with aspect ratio the geometric mean, as demonstrated
	in the image at top (14:9 is just slightly wider than the intersection). In this
	way 14:9 balances the needs of both 16:9 and 4:3, cropping or distorting both
	about equally. Similar considerations were used in the choice of 16:9 by the SMPTE,
	which balanced 2.35:1 and 4:3. </content>'
	- source_sentence: Solve the equation $7k^2 + 9k + 3 = d \cdot 7^a$ for positive
	integers $k$, $d$, and $a$, where $d < 7$.
	sentences:
	- <page_title> Rational square </page_title> <path> Square_numbers > Properties
	</path> <section_title> Properties </section_title> <content> Three squares are
	not sufficient for numbers of the form 4k(8m + 7). A positive integer can be represented
	as a sum of two squares precisely if its prime factorization contains no odd powers
	of primes of the form 4k + 3. This is generalized by Waring's problem. </content>
	- '<page_title> Personally Controlled Electronic Health Record </page_title> <path>
	Personally_Controlled_Electronic_Health_Record > Registration > Healthcare Identifiers
	Service (HI Service) </path> <section_title> Healthcare Identifiers Service (HI
	Service) </section_title> <content> The Healthcare Identifiers Service (HI Service)
	was established by the federal, state and territory governments to create unique
	identifiers for healthcare providers and individuals seeking healthcare. It was
	designed and implemented by Medicare Australia under the control of the NEHTA.
	The HI Service allocates three types of Healthcare Identifiers: Individual healthcare
	identifier (i.e., who received the service) The Individual Healthcare Identifier
	(IHI) is a unique 16 digit reference number that is used to identify individuals
	within the healthcare system. The healthcare provider can retrieve a registered
	patients IHI via the Healthcare Identifier Service by entering in the correct
	name, DOB, and Medicare number which will automatically retrieve the patients
	unique IHI from the system. </content>'
	- '<page_title> Systematic sampling </page_title> <path> Systematic_sampling </path>
	<section_title> Summary </section_title> <content> We want to give unit A a 20%
	probability of selection, unit B a 40% probability, and so on up to unit E (100%).
	Assuming we maintain alphabetical order, we allocate each unit to the following
	interval: A: 0 to 0.2 B: 0.2 to 0.6 (= 0.2 + 0.4) C: 0.6 to 1.2 (= 0.6 + 0.6)
	D: 1.2 to 2.0 (= 1.2 + 0.8) E: 2.0 to 3.0 (= 2.0 + 1.0) If our random start was
	0.156, we would first select the unit whose interval contains this number (i.e.
	A). Next, we would select the interval containing 1.156 (element C), then 2.156
	(element E). If instead our random start was 0.350, we would select from points
	0.350 (B), 1.350 (D), and 2.350 (E). </content>'
	- source_sentence: 'Given the linear transformation \( T: \mathbb{R}^3 \rightarrow
	\mathbb{R}^3 \) defined by $ T(x) = A(x) $ where \( A = \begin{pmatrix} 1 &
	1 & 1 \\ 0 & 1 & 2 \\ 1 & 2 & 2 \end{pmatrix} \), find the inverse transformation
	$ T^{-1}(x) $.'
	sentences:
	- <page_title> Genome sequence </page_title> <path> Genomic_sequence > Eukaryotic
	genomes </path> <section_title> Eukaryotic genomes </section_title> <content>
	In addition to the chromosomes in the nucleus, organelles such as the chloroplasts
	and mitochondria have their own DNA. Mitochondria are sometimes said to have their
	own genome often referred to as the "mitochondrial genome". The DNA found within
	the chloroplast may be referred to as the "plastome". </content>
	- <page_title> Right realism </page_title> <path> Right_realism > Overview > Rational
	choice theory </path> <section_title> Rational choice theory </section_title>
	<content> For example, in 1960 the steering columns of all cars in Germany were
	equipped with locks and the result was a 60 per cent reduction in car thefts.
	Whereas, in Great Britain only new cars were so equipped with the result being
	crime was displaced to the older unequipped cars. However, no evidence exists
	to suggest that an obscene phone caller will begin a career as a burglar. In response,
	Akers (1990) says that rational choice theorists make so many exceptions to the
	pure rationality stressed in their own models that nothing sets them apart from
	other theorists. Further, the rational choice models in literature have various
	situational or cognitive constraints and deterministic notions of cause and effect
	that render them, "...indistinguishable from current 'etiological' or 'positivist'
	theories." </content>
	- '<page_title> Elementary row operations </page_title> <path> Row_operations >
	Elementary row operations > Row-switching transformations > Properties </path>
	<section_title> Properties </section_title> <content> The inverse of this matrix
	is itself: T i , j − 1 = T i , j . {\displaystyle T_{i,j}^{-1}=T_{i,j}.} Since
	the determinant of the identity matrix is unity, det ( T i , j ) = − 1. {\displaystyle
	\det(T_{i,j})=-1.} </content>'
	- source_sentence: 'If \|x - 5\| = 23 what is the sum of all the values of x.

	A. A)46

	B. B)10

	C. C)56

	D. D)-46

	E. E)28'
	sentences:
	- <page_title> BMX racing </page_title> <path> BMX_racing > General rules of advancement
	in organized BMX racing > Professionals </path> <section_title> Professionals
	</section_title> <content> For example, if a rider participates in 13 national
	events, their best 10 will be considered and their worst three disregarded. This
	qualification must be met on the national level to wear National numbers one through
	ten on the number plate the following year. </content>
	- <page_title> Construction of the real numbers </page_title> <path> Constructions_of_real_numbers
	> Axiomatic definitions > Axioms > On models </path> <section_title> On models
	</section_title> <content> f(x +ℝ y) = f(x) +S f(y) and f(x ×ℝ y) = f(x) ×S f(y),
	for all x and y in R . {\displaystyle \mathbb {R} .} x ≤ℝ y if and only if f(x)
	≤S f(y), for all x and y in R . {\displaystyle \mathbb {R} .} </content>
	- <page_title> Rod calculus </page_title> <path> Rod_calculus > Subtraction > Without
	borrowing </path> <section_title> Without borrowing </section_title> <content>
	In situation in which no borrowing is needed, one only needs to take the number
	of rods in the subtrahend from the minuend. The result of the calculation is the
	difference. The adjacent image shows the steps in subtracting 23 from 54. </content>
	- source_sentence: For some constant $b$, if the minimum value of \[f(x)=\dfrac{x^2-2x+b}{x^2+2x+b}\]
	is $\tfrac12$, what is the maximum value of $f(x)$?
	sentences:
	- <page_title> Lagrangian multiplier </page_title> <path> Lagrange_multiplier >
	Examples > Example 1 </path> <section_title> Example 1 </section_title> <content>
	Evaluating the objective function f at these points yields f ( 2 2 , 2 2 ) = 2
	, f ( − 2 2 , − 2 2 ) = − 2 . {\displaystyle f\left({\tfrac {\sqrt {2\ }}{2}},{\tfrac
	{\sqrt {2\ }}{2}}\right)={\sqrt {2\ }}\ ,\qquad f\left(-{\tfrac {\sqrt {2\ }}{2}},-{\tfrac
	{\sqrt {2\ }}{2}}\right)=-{\sqrt {2\ }}~.} Thus the constrained maximum is 2 {\displaystyle
	\ {\sqrt {2\ }}\ } and the constrained minimum is − 2 {\displaystyle -{\sqrt {2}}}
	. </content>
	- '<page_title> Second degree polynomial </page_title> <path> Quadratic_function
	> Graph of the univariate function > Vertex > Maximum and minimum points </path>
	<section_title> Maximum and minimum points </section_title> <content> Using calculus,
	the vertex point, being a maximum or minimum of the function, can be obtained
	by finding the roots of the derivative: f ( x ) = a x 2 + b x + c ⇒ f ′ ( x )
	= 2 a x + b {\displaystyle f(x)=ax^{2}+bx+c\quad \Rightarrow \quad f''(x)=2ax+b}
	x is a root of f ''(x) if f ''(x) = 0 resulting in x = − b 2 a {\displaystyle
	x=-{\frac {b}{2a}}} with the corresponding function value f ( x ) = a ( − b 2
	a ) 2 + b ( − b 2 a ) + c = c − b 2 4 a , {\displaystyle f(x)=a\left(-{\frac {b}{2a}}\right)^{2}+b\left(-{\frac
	{b}{2a}}\right)+c=c-{\frac {b^{2}}{4a}},} so again the vertex point coordinates,
	(h, k), can be expressed as ( − b 2 a , c − b 2 4 a ) . {\displaystyle \left(-{\frac
	{b}{2a}},c-{\frac {b^{2}}{4a}}\right).} </content>'
	- '<page_title> Dimer model </page_title> <path> Domino_tiling > Counting tilings
	of regions </path> <section_title> Counting tilings of regions </section_title>
	<content> The number of ways to cover an m × n {\displaystyle m\times n} rectangle
	with m n 2 {\displaystyle {\frac {mn}{2}}} dominoes, calculated independently
	by Temperley & Fisher (1961) and Kasteleyn (1961), is given by (sequence A099390
	in the OEIS) When both m and n are odd, the formula correctly reduces to zero
	possible domino tilings. A special case occurs when tiling the 2 × n {\displaystyle
	2\times n} rectangle with n dominoes: the sequence reduces to the Fibonacci sequence.Another
	special case happens for squares with m = n = 0, 2, 4, 6, 8, 10, 12, ... is These
	numbers can be found by writing them as the Pfaffian of an m n × m n {\displaystyle
	mn\times mn} skew-symmetric matrix whose eigenvalues can be found explicitly.
	This technique may be applied in many mathematics-related subjects, for example,
	in the classical, 2-dimensional computation of the dimer-dimer correlator function
	in statistical mechanics. The number of tilings of a region is very sensitive
	to boundary conditions, and can change dramatically with apparently insignificant
	changes in the shape of the region. This is illustrated by the number of tilings
	of an Aztec diamond of order n, where the number of tilings is 2(n + 1)n/2. If
	this is replaced by the "augmented Aztec diamond" of order n with 3 long rows
	in the middle rather than 2, the number of tilings drops to the much smaller number
	D(n,n), a Delannoy number, which has only exponential rather than super-exponential
	growth in n. For the "reduced Aztec diamond" of order n with only one long middle
	row, there is only one tiling. </content>'
	pipeline_tag: sentence-similarity
	library_name: sentence-transformers
	---

	# SentenceTransformer based on sentence-transformers/all-MiniLM-L6-v2

	This is a [sentence-transformers](https://www.SBERT.net) model finetuned from [sentence-transformers/all-MiniLM-L6-v2](https://huggingface.co/sentence-transformers/all-MiniLM-L6-v2). It maps sentences & paragraphs to a 384-dimensional dense vector space and can be used for semantic textual similarity, semantic search, paraphrase mining, text classification, clustering, and more.

	## Model Details

	### Model Description
	- Model Type: Sentence Transformer
	- Base model: [sentence-transformers/all-MiniLM-L6-v2](https://huggingface.co/sentence-transformers/all-MiniLM-L6-v2) <!-- at revision c9745ed1d9f207416be6d2e6f8de32d1f16199bf -->
	- Maximum Sequence Length: 384 tokens
	- Output Dimensionality: 384 dimensions
	- Similarity Function: Cosine Similarity
	<!-- - Training Dataset: Unknown -->
	<!-- - Language: Unknown -->
	<!-- - License: Unknown -->

	### Model Sources

	- Documentation: [Sentence Transformers Documentation](https://sbert.net)
	- Repository: [Sentence Transformers on GitHub](https://github.com/UKPLab/sentence-transformers)
	- Hugging Face: [Sentence Transformers on Hugging Face](https://huggingface.co/models?library=sentence-transformers)

	### Full Model Architecture

	```
	SentenceTransformer(
	(0): Transformer({'max_seq_length': 384, 'do_lower_case': False}) with Transformer model: BertModel
	(1): Pooling({'word_embedding_dimension': 384, 'pooling_mode_cls_token': False, 'pooling_mode_mean_tokens': True, 'pooling_mode_max_tokens': False, 'pooling_mode_mean_sqrt_len_tokens': False, 'pooling_mode_weightedmean_tokens': False, 'pooling_mode_lasttoken': False, 'include_prompt': True})
	(2): Normalize()
	)
	```

	## Usage

	### Direct Usage (Sentence Transformers)

	First install the Sentence Transformers library:

	```bash
	pip install -U sentence-transformers
	```

	Then you can load this model and run inference.
	```python
	from sentence_transformers import SentenceTransformer

	# Download from the 🤗 Hub
	model = SentenceTransformer("Lysandrec/MNLP_M2_document_encoder")
	# Run inference
	sentences = [
	'For some constant $b$, if the minimum value of \\[f(x)=\\dfrac{x^2-2x+b}{x^2+2x+b}\\] is $\\tfrac12$, what is the maximum value of $f(x)$?',
	"<page_title> Second degree polynomial </page_title> <path> Quadratic_function > Graph of the univariate function > Vertex > Maximum and minimum points </path> <section_title> Maximum and minimum points </section_title> <content> Using calculus, the vertex point, being a maximum or minimum of the function, can be obtained by finding the roots of the derivative: f ( x ) = a x 2 + b x + c ⇒ f ′ ( x ) = 2 a x + b {\\displaystyle f(x)=ax^{2}+bx+c\\quad \\Rightarrow \\quad f'(x)=2ax+b} x is a root of f '(x) if f '(x) = 0 resulting in x = − b 2 a {\\displaystyle x=-{\\frac {b}{2a}}} with the corresponding function value f ( x ) = a ( − b 2 a ) 2 + b ( − b 2 a ) + c = c − b 2 4 a , {\\displaystyle f(x)=a\\left(-{\\frac {b}{2a}}\\right)^{2}+b\\left(-{\\frac {b}{2a}}\\right)+c=c-{\\frac {b^{2}}{4a}},} so again the vertex point coordinates, (h, k), can be expressed as ( − b 2 a , c − b 2 4 a ) . {\\displaystyle \\left(-{\\frac {b}{2a}},c-{\\frac {b^{2}}{4a}}\\right).} </content>",
	'<page_title> Lagrangian multiplier </page_title> <path> Lagrange_multiplier > Examples > Example 1 </path> <section_title> Example 1 </section_title> <content> Evaluating the objective function f at these points yields f ( 2 2 , 2 2 ) = 2 , f ( − 2 2 , − 2 2 ) = − 2 . {\\displaystyle f\\left({\\tfrac {\\sqrt {2\\ }}{2}},{\\tfrac {\\sqrt {2\\ }}{2}}\\right)={\\sqrt {2\\ }}\\ ,\\qquad f\\left(-{\\tfrac {\\sqrt {2\\ }}{2}},-{\\tfrac {\\sqrt {2\\ }}{2}}\\right)=-{\\sqrt {2\\ }}~.} Thus the constrained maximum is 2 {\\displaystyle \\ {\\sqrt {2\\ }}\\ } and the constrained minimum is − 2 {\\displaystyle -{\\sqrt {2}}} . </content>',
	]
	embeddings = model.encode(sentences)
	print(embeddings.shape)
	# [3, 384]

	# Get the similarity scores for the embeddings
	similarities = model.similarity(embeddings, embeddings)
	print(similarities.shape)
	# [3, 3]
	```

	<!--
	### Direct Usage (Transformers)

	<details><summary>Click to see the direct usage in Transformers</summary>

	</details>
	-->

	<!--
	### Downstream Usage (Sentence Transformers)

	You can finetune this model on your own dataset.

	<details><summary>Click to expand</summary>

	</details>
	-->

	<!--
	### Out-of-Scope Use

	List how the model may foreseeably be misused and address what users ought not to do with the model.
	-->

	<!--
	## Bias, Risks and Limitations

	What are the known or foreseeable issues stemming from this model? You could also flag here known failure cases or weaknesses of the model.
	-->

	<!--
	### Recommendations

	What are recommendations with respect to the foreseeable issues? For example, filtering explicit content.
	-->

	## Training Details

	### Training Dataset

	#### Unnamed Dataset

	This training dataset was synthetically generated. For each question from the source Q/A dataset (`Lysandrec/MNLP_M2_rag_dataset`), relevant passages were retrieved from a large document corpus (`Lysandrec/MNLP_M2_rag_documents`).
	- A positive_passage was identified from the retrieved candidates, typically one containing the answer to the question. If no definitive positive was found, the top retrieved passage was often selected.
	- Hard_negative_passages were selected from other highly-ranked (but not positive) retrieved documents for the same question.
	- Random_negative_passages were sampled from the broader document corpus, ensuring they differed from the selected positive and hard negative passages.
	This process resulted in triplets of (query, positive_passage, negative_passage) used for training.

	* Size: 100,000 training samples
	* Columns: `query` (a question), `positive_passage` (a good retrieved document), and `negative_passage` (a bad example of a retrieved document)
	* Approximate statistics based on the first 1000 samples:
	\| \| query \| positive_passage \| negative_passage \|
	\|:--------\|:------------------------------------------------------------------------------------\|:-------------------------------------------------------------------------------------\|:------------------------------------------------------------------------------------\|
	\| type \| string \| string \| string \|
	\| details \| <ul><li>min: 13 tokens</li><li>mean: 64.12 tokens</li><li>max: 214 tokens</li></ul> \| <ul><li>min: 64 tokens</li><li>mean: 205.95 tokens</li><li>max: 384 tokens</li></ul> \| <ul><li>min: 63 tokens</li><li>mean: 178.5 tokens</li><li>max: 384 tokens</li></ul> \|
	* Samples:
	\| query \| positive_passage \| negative_passage \|
	\|:----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\|:------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\|:---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------\|
	\| `The average of first five prime numbers greater than 61 is?<br>A. A)32.2<br>B. B)32.98<br>C. C)74.6<br>D. D)32.8<br>E. E)32.4` \| `<page_title> 61 (number) </page_title> <path> 61_(number) > In mathematics </path> <section_title> In mathematics </section_title> <content> 61 is: the 18th prime number. a twin prime with 59. a cuban prime of the form p = x3 − y3/x − y, where x = y + 1. the smallest proper prime, a prime p which ends in the digit 1 in base 10 and whose reciprocal in base 10 has a repeating sequence with length p − 1. In such primes, each digit 0, 1, ..., 9 appears in the repeating sequence the same number of times as does each other digit (namely, p − 1/10 times). </content>` \| `<page_title> Astatine </page_title> <path> Element_85 > Characteristics > Chemical </path> <section_title> Chemical </section_title> <content> In comparison, the value of Cl (349) is 6.4% higher than F (328); Br (325) is 6.9% less than Cl; and I (295) is 9.2% less than Br. The marked reduction for At was predicted as being due to spin–orbit interactions. The first ionization energy of astatine is about 899 kJ mol−1, which continues the trend of decreasing first ionization energies down the halogen group (fluorine, 1681; chlorine, 1251; bromine, 1140; iodine, 1008). </content>` \|
	\| `A charitable association sold an average of 66 raffle tickets per member. Among the female members, the average was 70 raffle tickets. The male to female ratio of the association is 1:2. What was the average number E of tickets sold by the male members of the association<br>A. A)50<br>B. B)56<br>C. C)58<br>D. D)62<br>E. E)66` \| `<page_title> RSA number </page_title> <path> RSA_numbers </path> <section_title> Summary </section_title> <content> Cash prizes of varying size, up to US$200,000 (and prizes up to $20,000 awarded), were offered for factorization of some of them. The smallest RSA number was factored in a few days. Most of the numbers have still not been factored and many of them are expected to remain unfactored for many years to come. </content>` \| `<page_title> Peer learning </page_title> <path> Peer_learning > Connections with other practices > Connectivism </path> <section_title> Connectivism </section_title> <content> Yochai Benkler explains how the now-ubiquitous computer helps us produce and process knowledge together with others in his book, The Wealth of Networks. George Siemens argues in Connectivism: A Learning Theory for the Digital Age, that technology has changed the way we learn, explaining how it tends to complicate or expose the limitations of the learning theories of the past. In practice, the ideas of connectivism developed in and alongside the then-new social formation, "massive open online courses" or MOOCs. Connectivism proposes that the knowledge we can access by virtue of our connections with others is just as valuable as the information carried inside our minds. </content>` \|
	\| `Find prime numbers $a, b, c, d, e$ such that $a^4 + b^4 + c^4 + d^4 + e^4 = abcde$.` \| `<page_title> Pythagorean triangle </page_title> <path> Primitive_Pythagorean_triple > Special cases and related equations > The Jacobi–Madden equation </path> <section_title> The Jacobi–Madden equation </section_title> <content> The equation, a 4 + b 4 + c 4 + d 4 = ( a + b + c + d ) 4 {\displaystyle a^{4}+b^{4}+c^{4}+d^{4}=(a+b+c+d)^{4}} is equivalent to the special Pythagorean triple, ( a 2 + a b + b 2 ) 2 + ( c 2 + c d + d 2 ) 2 = ( ( a + b ) 2 + ( a + b ) ( c + d ) + ( c + d ) 2 ) 2 {\displaystyle (a^{2}+ab+b^{2})^{2}+(c^{2}+cd+d^{2})^{2}=((a+b)^{2}+(a+b)(c+d)+(c+d)^{2})^{2}} There is an infinite number of solutions to this equation as solving for the variables involves an elliptic curve. Small ones are, a , b , c , d = − 2634 , 955 , 1770 , 5400 {\displaystyle a,b,c,d=-2634,955,1770,5400} a , b , c , d = − 31764 , 7590 , 27385 , 48150 {\displaystyle a,b,c,d=-31764,7590,27385,48150} </content>` \| `<page_title> Pythagorean triple </page_title> <path> Pythagorean_triples > Special cases and related equations > Descartes' Circle Theorem </path> <section_title> Descartes' Circle Theorem </section_title> <content> For the case of Descartes' circle theorem where all variables are squares, 2 ( a 4 + b 4 + c 4 + d 4 ) = ( a 2 + b 2 + c 2 + d 2 ) 2 {\displaystyle 2(a^{4}+b^{4}+c^{4}+d^{4})=(a^{2}+b^{2}+c^{2}+d^{2})^{2}} Euler showed this is equivalent to three simultaneous Pythagorean triples, ( 2 a b ) 2 + ( 2 c d ) 2 = ( a 2 + b 2 − c 2 − d 2 ) 2 {\displaystyle (2ab)^{2}+(2cd)^{2}=(a^{2}+b^{2}-c^{2}-d^{2})^{2}} ( 2 a c ) 2 + ( 2 b d ) 2 = ( a 2 − b 2 + c 2 − d 2 ) 2 {\displaystyle (2ac)^{2}+(2bd)^{2}=(a^{2}-b^{2}+c^{2}-d^{2})^{2}} ( 2 a d ) 2 + ( 2 b c ) 2 = ( a 2 − b 2 − c 2 + d 2 ) 2 {\displaystyle (2ad)^{2}+(2bc)^{2}=(a^{2}-b^{2}-c^{2}+d^{2})^{2}} There is also an infinite number of solutions, and for the special case when a + b = c {\displaystyle a+b=c} , then the equation simplifi...</code> \|
	* Loss: [`TripletLoss`](https://sbert.net/docs/package_reference/sentence_transformer/losses.html#tripletloss) with these parameters:
	```json
	{
	"distance_metric": "TripletDistanceMetric.EUCLIDEAN",
	"triplet_margin": 5
	}
	```

	### Training Hyperparameters
	#### Non-Default Hyperparameters

	- `per_device_train_batch_size`: 64
	- `per_device_eval_batch_size`: 64
	- `num_train_epochs`: 1
	- `multi_dataset_batch_sampler`: round_robin

	#### All Hyperparameters
	<details><summary>Click to expand</summary>

	- `overwrite_output_dir`: False
	- `do_predict`: False
	- `eval_strategy`: no
	- `prediction_loss_only`: True
	- `per_device_train_batch_size`: 64
	- `per_device_eval_batch_size`: 64
	- `per_gpu_train_batch_size`: None
	- `per_gpu_eval_batch_size`: None
	- `gradient_accumulation_steps`: 1
	- `eval_accumulation_steps`: None
	- `torch_empty_cache_steps`: None
	- `learning_rate`: 5e-05
	- `weight_decay`: 0.0
	- `adam_beta1`: 0.9
	- `adam_beta2`: 0.999
	- `adam_epsilon`: 1e-08
	- `max_grad_norm`: 1
	- `num_train_epochs`: 1
	- `max_steps`: -1
	- `lr_scheduler_type`: linear
	- `lr_scheduler_kwargs`: {}
	- `warmup_ratio`: 0.0
	- `warmup_steps`: 0
	- `log_level`: passive
	- `log_level_replica`: warning
	- `log_on_each_node`: True
	- `logging_nan_inf_filter`: True
	- `save_safetensors`: True
	- `save_on_each_node`: False
	- `save_only_model`: False
	- `restore_callback_states_from_checkpoint`: False
	- `no_cuda`: False
	- `use_cpu`: False
	- `use_mps_device`: False
	- `seed`: 42
	- `data_seed`: None
	- `jit_mode_eval`: False
	- `use_ipex`: False
	- `bf16`: False
	- `fp16`: False
	- `fp16_opt_level`: O1
	- `half_precision_backend`: auto
	- `bf16_full_eval`: False
	- `fp16_full_eval`: False
	- `tf32`: None
	- `local_rank`: 0
	- `ddp_backend`: None
	- `tpu_num_cores`: None
	- `tpu_metrics_debug`: False
	- `debug`: []
	- `dataloader_drop_last`: False
	- `dataloader_num_workers`: 0
	- `dataloader_prefetch_factor`: None
	- `past_index`: -1
	- `disable_tqdm`: False
	- `remove_unused_columns`: True
	- `label_names`: None
	- `load_best_model_at_end`: False
	- `ignore_data_skip`: False
	- `fsdp`: []
	- `fsdp_min_num_params`: 0
	- `fsdp_config`: {'min_num_params': 0, 'xla': False, 'xla_fsdp_v2': False, 'xla_fsdp_grad_ckpt': False}
	- `tp_size`: 0
	- `fsdp_transformer_layer_cls_to_wrap`: None
	- `accelerator_config`: {'split_batches': False, 'dispatch_batches': None, 'even_batches': True, 'use_seedable_sampler': True, 'non_blocking': False, 'gradient_accumulation_kwargs': None}
	- `deepspeed`: None
	- `label_smoothing_factor`: 0.0
	- `optim`: adamw_torch
	- `optim_args`: None
	- `adafactor`: False
	- `group_by_length`: False
	- `length_column_name`: length
	- `ddp_find_unused_parameters`: None
	- `ddp_bucket_cap_mb`: None
	- `ddp_broadcast_buffers`: False
	- `dataloader_pin_memory`: True
	- `dataloader_persistent_workers`: False
	- `skip_memory_metrics`: True
	- `use_legacy_prediction_loop`: False
	- `push_to_hub`: False
	- `resume_from_checkpoint`: None
	- `hub_model_id`: None
	- `hub_strategy`: every_save
	- `hub_private_repo`: None
	- `hub_always_push`: False
	- `gradient_checkpointing`: False
	- `gradient_checkpointing_kwargs`: None
	- `include_inputs_for_metrics`: False
	- `include_for_metrics`: []
	- `eval_do_concat_batches`: True
	- `fp16_backend`: auto
	- `push_to_hub_model_id`: None
	- `push_to_hub_organization`: None
	- `mp_parameters`:
	- `auto_find_batch_size`: False
	- `full_determinism`: False
	- `torchdynamo`: None
	- `ray_scope`: last
	- `ddp_timeout`: 1800
	- `torch_compile`: False
	- `torch_compile_backend`: None
	- `torch_compile_mode`: None
	- `include_tokens_per_second`: False
	- `include_num_input_tokens_seen`: False
	- `neftune_noise_alpha`: None
	- `optim_target_modules`: None
	- `batch_eval_metrics`: False
	- `eval_on_start`: False
	- `use_liger_kernel`: False
	- `eval_use_gather_object`: False
	- `average_tokens_across_devices`: False
	- `prompts`: None
	- `batch_sampler`: batch_sampler
	- `multi_dataset_batch_sampler`: round_robin

	</details>

	### Training Logs
	\| Epoch \| Step \| Training Loss \|
	\|:------:\|:----:\|:-------------:\|
	\| 0.3199 \| 500 \| 4.0855 \|
	\| 0.6398 \| 1000 \| 3.9274 \|
	\| 0.9597 \| 1500 \| 3.9199 \|


	### Framework Versions
	- Python: 3.12.8
	- Sentence Transformers: 3.4.1
	- Transformers: 4.51.3
	- PyTorch: 2.5.1+cu124
	- Accelerate: 1.3.0
	- Datasets: 3.2.0
	- Tokenizers: 0.21.0

	## Citation

	### BibTeX

	#### Sentence Transformers
	```bibtex
	@inproceedings{reimers-2019-sentence-bert,
	title = "Sentence-BERT: Sentence Embeddings using Siamese BERT-Networks",
	author = "Reimers, Nils and Gurevych, Iryna",
	booktitle = "Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing",
	month = "11",
	year = "2019",
	publisher = "Association for Computational Linguistics",
	url = "https://arxiv.org/abs/1908.10084",
	}
	```

	#### TripletLoss
	```bibtex
	@misc{hermans2017defense,
	title={In Defense of the Triplet Loss for Person Re-Identification},
	author={Alexander Hermans and Lucas Beyer and Bastian Leibe},
	year={2017},
	eprint={1703.07737},
	archivePrefix={arXiv},
	primaryClass={cs.CV}
	}
	```

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