README.md

metadata

language:
  - en
license: apache-2.0
tags:
  - sentence-transformers
  - sentence-similarity
  - feature-extraction
  - generated_from_trainer
  - dataset_size:1140
  - loss:MatryoshkaLoss
  - loss:MultipleNegativesRankingLoss
base_model: NovaSearch/stella_en_400M_v5
widget:
  - source_sentence: Prove that x + |x - 7| ≥ 7
    sentences:
      - >-
        The subtyping relationship between `Iterable[Pair[A, Y]]` and `Map[A,
        Y]` can be understood by examining the covariance and structure of these
        types.


        1. **Covariance**: Both `Iterable` and `Pair` are covariant in their
        type parameters. This means that if `A` is a supertype of `B`, then
        `Iterable[Pair[A, Y]]` can be treated as a subtype of `Iterable[Pair[B,
        Y]]`.


        2. **Map's Structure**: The `Map` class extends `Iterable[Pair[U, V]]`,
        where `U` is invariant and `V` is covariant. Therefore, `Map[A, Y]` is
        considered an `Iterable[Pair[A, Y]]`. 


        3. **Type Compatibility**: Since `Map[A, Y]` is an `Iterable[Pair[A,
        Y]]`, it follows that any instance of `Map[A, Y]` can be treated as an
        instance of `Iterable[Pair[A, Y]]`. 


        4. **Subtyping**: Given that `Iterable[Pair[A, Y]]` can be seen as an
        interface that describes a broader category of collections, while
        `Map[A, Y]` is a specific implementation of that interface, we conclude
        that `Iterable[Pair[A, Y]]` is a supertype of `Map[A, Y]`. Thus,
        `Iterable[Pair[A, Y]]` can be assigned to a variable of type `Map[A,
        Y]`, making the relationship `Iterable[Pair[A, Y]] <: Map[A, Y]`.


        Therefore, the correct subtyping relationship is `Iterable[Pair[A, Y]]
        <: Map[A, Y]`.
      - >-
        To implement a function that inserts an element into a sorted list while
        maintaining the sorted order, we can use recursion. The idea is to check
        the first element of the list and decide whether to insert the new
        element before it or to continue the recursion with the rest of the
        list.


        Here’s how you can implement the `insert` function in Scala:


        ```scala

        def insert(elem: Int, list: List[Int]): List[Int] = list match {
          case Nil => List(elem) // If the list is empty, return a new list with the element
          case head :: tail =>
            if (elem <= head) {
              elem :: list // If the element is less than or equal to the head, insert it at the front
            } else {
              head :: insert(elem, tail) // Otherwise, keep the head and recurse on the tail
            }
        }

        ```


        ### Explanation:

        1. **Base Case**: 
           - If the list is empty (`Nil`), we return a new list that contains only the element `elem`.
           
        2. **Recursive Case**:
           - The list is matched against its head and tail using pattern matching. `head` is the first element, and `tail` is the remainder of the list.
           - If `elem` is less than or equal to `head`, we prepend `elem` to the list (using `::`), resulting in a new list that maintains the sorted order.
           - If `elem` is greater than `head`, we keep `head` in the resulting list and recursively call `insert` with `elem` and `tail`. This continues until we find the correct position for `elem`.

        ### Example Usage:

        ```scala

        val sortedList = List(1, 3, 5, 7)

        val newElement = 4

        val newList = insert(newElement, sortedList)

        // newList will be List(1, 3, 4, 5, 7)

        ```


        This recursive approach effectively ensures that the newly inserted
        element maintains the sorted property of the list.
      - >-
        To prove the inequality \( x + |x - 7| \geq 7 \), we will analyze it by
        considering two cases based on the definition of the absolute value
        function.


        ### Step 1: Understanding the Absolute Value


        The absolute value function \( |x - 7| \) can be expressed in terms of
        piecewise functions:


        \[

        |x - 7| = 

        \begin{cases} 

        7 - x & \text{if } x < 7 \\

        x - 7 & \text{if } x \geq 7 

        \end{cases}

        \]


        ### Step 2: Case 1: \( x < 7 \)


        In this case, we have:


        \[

        |x - 7| = 7 - x

        \]


        Substituting this into the inequality \( x + |x - 7| \):


        \[

        x + |x - 7| = x + (7 - x) = 7

        \]


        Now, we need to check if:


        \[

        7 \geq 7

        \]


        This statement is true. Therefore, for all \( x < 7 \), the inequality
        \( x + |x - 7| \geq 7 \) holds.


        ### Step 3: Case 2: \( x \geq 7 \)


        In this case, we have:


        \[

        |x - 7| = x - 7

        \]


        Substituting this into the inequality \( x + |x - 7| \):


        \[

        x + |x - 7| = x + (x - 7) = 2x - 7

        \]


        Now, we need to check if:


        \[

        2x - 7 \geq 7

        \]


        To solve this inequality, we can rearrange it:


        \[

        2x \geq 14 \\

        x \geq 7

        \]


        This statement is also true for all \( x \geq 7 \).


        ### Conclusion


        Combining both cases, we find that the inequality \( x + |x - 7| \geq 7
        \) holds true for all \( x \in \mathbb{R} \).


        Thus, we have proven that:


        \[

        x + |x - 7| \geq 7 \quad \forall x \in \mathbb{R}.

        \]
  - source_sentence: >-
      Assume you are working on SuperQuiz, a trendy app that lets everyone
      design quizzes and share them with friends! SuperQuiz recently hired a new
      CEO, who wants to improve the development practices using modern methods.
      However, this CEO has no engineering background, so the suggested
      improvements are well intentioned but not always feasible. The latest CEO
      suggestion is this:

      "Continuous Integration is a modern best practice. We must adopt it, so
      that the code in our repository never has bugs. From now on, all branches
      in the SuperQuiz repository must have continuous integration enabled, and
      at the end of each day all branches must pass all tests."

      Propose (in 1-2 sentences) a compromise that achieves the CEO's true
      objective:
    sentences:
      - >-
        To effectively implement continuous integration while maintaining
        practicality, we should set it up for the main branch of the SuperQuiz
        repository. This approach will help ensure that the code merged into the
        main branch is thoroughly tested, thereby reducing the likelihood of
        bugs in the final product while allowing developers to work on feature
        branches without the burden of daily testing requirements.
      - >-
        When removing a method from a project due to its potential for misuse
        and replacing it with a more user-friendly alternative, consider the
        following structured steps for upcoming releases:


        ### 1. **Evaluate the Impact of Removal**
           - **Dependency Analysis**: Identify all components, modules, or external libraries that depend on the method.
           - **Usage Tracking**: Analyze how frequently the method is used across the codebase and document its current usage patterns.

        ### 2. **Design the Replacement**
           - **Feature Comparison**: Ensure the new method covers all functionalities of the old method while being easier to use.
           - **User Experience**: Focus on enhancing usability and reducing the likelihood of errors with the new method.

        ### 3. **Update Documentation**
           - **API Documentation**: Revise the API documentation to remove references to the old method and provide comprehensive details about the new method.
           - **Migration Guide**: Create a clear migration guide outlining how to transition from the old method to the new one.

        ### 4. **Deprecate the Old Method**
           - **Deprecation Notice**: Mark the old method as deprecated, ensuring it remains available for a defined period while encouraging users to transition.
           - **Warnings**: Implement warnings in the code to alert developers using the old method about its deprecation.

        ### 5. **Refactor Codebase**
           - **Replace Instances**: Update all instances of the old method in the codebase with the new method, ensuring that functionality remains intact.
           - **Testing**: Write and run unit tests to verify that the new method behaves as expected and does not introduce new bugs.

        ### 6. **Communicate Changes**
           - **Release Notes**: Include detailed information about the removal and replacement in the release notes for users.
           - **Community Announcement**: If applicable, communicate the changes through relevant channels (e.g., newsletters, forums, or social media).

        ### 7. **Monitor Adoption**
           - **Feedback Mechanism**: Establish a way for users to provide feedback on the new method, allowing for quick identification of any issues.
           - **Usage Metrics**: Track the adoption rate of the new method versus the old one to gauge the success of the transition.

        ### 8. **Plan for Future Releases**
           - **Versioning Strategy**: Decide on a versioning strategy to handle the removal (e.g., major version bump) and inform users of the changes.
           - **Continuous Improvement**: Gather feedback post-release and refine the new method as needed based on user experience.

        By following these steps, you can ensure a smooth transition from the
        old method to the new one while minimizing disruption for users and
        maintaining the integrity of the project.
      - >-
        Draft: 


        To solve the problem of finding a non-negative vertex potential \( p(v)
        \) for each vertex \( v \in V \) in a graph \( G = (V, E) \), we can
        formulate it as a linear program and utilize a polynomial-time
        separation oracle to verify feasibility. Below are the detailed steps to
        achieve this.


        ### Step 1: Formulate the Linear Program

        We will set up a linear program with variables corresponding to the
        potentials of each vertex:


        - **Objective Function**:
          \[
          \text{Maximize } Z = \sum_{v \in V} p(v)
          \]

        - **Constraints**:
          For every non-empty subset \( S \subset V \):
          \[
          \sum_{v \in S} p(v) \leq |E(S, \bar{S})|
          \]

        - **Non-negativity Constraints**:
          \[
          p(v) \geq 0 \quad \text{for all } v \in V
          \]

        ### Step 2: Design a Separation Oracle

        To check whether a given potential vector \( p^* \) is feasible, we need
        to verify the constraints efficiently. We will construct a function \(
        f(S) \) for each non-empty subset \( S \subset V \):

        \[

        f(S) = |E(S, \bar{S})| - \sum_{v \in S} p^*(v)

        \]


        Now, we need to check if \( f(S) \geq 0 \) holds for all \( \emptyset
        \subset S \subset V \). If \( \min_{\emptyset \subseteq S \subset V}
        f(S) < 0 \), then \( p^* \) is not feasible, and we can output a
        violated constraint.


        ### Step 3: Submodularity of \( f(S) \)

        The function \( f(S) \) is a submodular function as it is the sum of a
        cut function (which is submodular) and a linear term (which is trivially
        submodular). The key insight is that identifying violated constraints is
        equivalent to minimizing the submodular function \( f(S) \).


        ### Step 4: Solve the Submodular Function Minimization

        Since we do not allow \( S = V \) in our solutions, we can solve the
        submodular minimization for the smaller ground sets \( V \setminus
        \{v_1\}, V \setminus \{v_2\}, \ldots, V \setminus \{v_n\} \). Each of
        these instances can be solved using polynomial-time algorithms for
        submodular function minimization.


        ### Conclusion

        By constructing a feasible region using the LP and developing a
        separation oracle that confirms whether a given potential is feasible or
        outputs a violated constraint, we can solve the original problem in
        polynomial time using the Ellipsoid method.


        Answer:


        To design a polynomial-time algorithm for the problem of finding a
        vertex potential \( p(v) \) in a graph \( G = (V, E) \), we can use
        linear programming combined with a separation oracle:


        1. **Linear Program Formulation**:
           We define the following linear program:
           - **Objective**:
             \[
             \text{Maximize } Z = \sum_{v \in V} p(v)
             \]
           - **Constraints**:
             For every non-empty subset \( S \subset V \):
             \[
             \sum_{v \in S} p(v) \leq |E(S, \bar{S})|
             \]
           - **Non-negativity**:
             \[
             p(v) \geq 0 \quad \text{for all } v \in V
             \]

        2. **Separation Oracle**:
           To check if a potential vector \( p^* \) is feasible, we define:
           \[
           f(S) = |E(S, \bar{S})| - \sum_{v \in S} p^*(v)
           \]
           We need to determine if \( \min_{\emptyset \subseteq S \subset V} f(S) < 0 \). If it is, \( p^* \) is not feasible.

        3. **Submodular Function**:
           The function \( f(S) \) is submodular, and we can solve the minimization problem for the smaller sets \( V \setminus \{v_1\}, V \setminus \{v_2\}, \ldots, V \setminus \{v_n\} \) to find any violations efficiently.

        4. **Polynomial Time**:
           The submodular function minimization can be solved in polynomial time, enabling us to check feasibility of \( p^* \) efficiently. As a result, this method allows us to apply the Ellipsoid method to solve the linear program in polynomial time.

        Justification:


        The approach leverages linear programming to maximize the vertex
        potentials while adhering to the specified constraints. By formulating
        the problem in terms of a separation oracle, we can efficiently verify
        the feasibility of potential solutions. The use of submodular functions
        is crucial because they have well-established properties that allow us
        to find violations quickly. This guarantees that the algorithm runs in
        polynomial time due to the efficiency of the submodular minimization
        process. Overall, this method combines theoretical insights from
        combinatorial optimization with practical computational techniques,
        leading to a robust solution to the problem.
  - source_sentence: >-
      Let $A \in \mathbb{R}^{m\times n}$, $b\in \mathbb{R}^m$ and $c\in
      \mathbb{R}^n$. Consider the following linear program with $n$ variables:
      \begin{align*} \textbf{maximize} \hspace{0.8cm} & c^Tx \\ \textbf{subject
      to}\hspace{0.8cm} & Ax =b \\ \hspace{0.8cm} & x \geq 0 \end{align*} Show
      that any extreme point $x^*$ has at most $m$ non-zero entries, i.e.,
      $|\{i: x^*_i > 0 \}| \leq m$. \\[-0.2cm] \noindent \emph{Hint: what
      happens if the columns corresponding to non-zero entries in $x^*$ are
      linearly dependent?}\\[-0.2cm] {\small (If you are in a good mood you can
      prove the following stronger statement: $x^*$ is an extreme point if and
      only if the columns of $A$ corresponding to non-zero entries of $x^*$ are
      linearly independent.)}
    sentences:
      - >-
        To show that any extreme point \( x^* \) of the given linear program has
        at most \( m \) non-zero entries, we proceed with the following steps:


        1. **Definition of Extreme Points**: An extreme point \( x^* \) is one
        that cannot be represented as a convex combination of other feasible
        points. In the context of the linear program, this means that if \( x^*
        \) has more than \( m \) non-zero entries, it can potentially be
        expressed as such a combination.


        2. **Identifying Non-Zero Columns**: Let \( S = \{ i : x_i^* > 0 \} \).
        If \( |S| > m \), then we have more than \( m \) non-zero variables in
        \( x^* \). The corresponding columns of matrix \( A \) associated with
        these non-zero entries are denoted as \( A_S \).


        3. **Linear Dependence**: Since there are more than \( m \) columns in
        \( A_S \), and given that the maximum rank of any set of vectors
        (columns) is limited by the number of rows (which is \( m \)), the
        columns of \( A_S \) must be linearly dependent when \( |S| > m \).


        4. **Feasibility Condition**: The linear dependence implies that we can
        find a non-trivial combination of these columns that sums to zero,
        allowing us to perturb some components of \( x^* \) while still
        satisfying the equality constraint \( Ax = b \). This contradicts the
        property of extreme points.


        5. **Conclusion**: Therefore, if an extreme point has more than \( m \)
        non-zero entries, it cannot be an extreme point due to linear dependence
        among its corresponding columns in \( A \). Thus, we conclude that any
        extreme point \( x^* \) must have at most \( m \) non-zero entries,
        i.e., \( |\{i: x_i^* > 0\}| \leq m \).
      - >-
        The sentences produced by the recognizer, such as "A was salmon outer
        the does" and "I Thomas at mice not the spoon," exhibit a lack of
        grammatical coherence and semantic meaning. The issues with these
        sentences can be categorized as follows:


        1. **Grammatical Errors**: The sentences contain phrases that do not
        follow the rules of English grammar. For example, "A was salmon outer
        the does" lacks a clear subject-verb-object structure, and "I Thomas at
        mice not the spoon" is not a coherent expression.


        2. **Semantic Incoherence**: Even if the sentences were grammatically
        correct, they still do not convey meaningful ideas. Words are used in
        ways that do not align with their typical meanings or relations, leading
        to nonsensical phrases.


        To improve the quality of the recognizer and select the correct
        sequences of words, the following Natural Language Processing (NLP)
        techniques can be employed:


        1. **Language Modeling**: Utilizing statistical or neural language
        models (e.g., n-grams, recurrent neural networks, transformers) can help
        evaluate the probability of word sequences. These models can predict the
        likelihood of a sequence of words based on the context and structure of
        the language.


        2. **Syntactic Parsing**: Implementing syntactic parsers can help
        identify the grammatical structure of sentences. This will allow the
        system to filter out those that do not conform to acceptable grammatical
        rules.


        3. **Semantic Analysis**: Techniques such as word embeddings (e.g.,
        Word2Vec, GloVe) or transformer-based models (e.g., BERT) can be used to
        assess the semantic similarity of phrases or to identify word meanings
        in context. This can help in selecting sentences that are not only
        grammatically correct but also semantically meaningful.


        4. **Contextual Constraints**: Incorporating contextual information from
        the surrounding text or the specific domain of the input can help guide
        the selection process, ensuring that the output is relevant and
        coherent.


        ### Required Resources:

        - **Training Data**: A large corpus of correctly structured and
        meaningful English sentences for training language models and parsers.

        - **Computational Power**: Access to sufficient computational resources
        (e.g., GPUs) for training deep learning models, especially for complex
        transformer architectures.

        - **NLP Libraries and Frameworks**: Utilizing libraries such as NLTK,
        spaCy, or Hugging Face Transformers for implementing various NLP
        techniques.

        - **Expertise in NLP**: Knowledge of linguistic principles, machine
        learning, and natural language processing methodologies will be
        essential to design and implement an effective solution.


        By applying these techniques and leveraging the necessary resources, the
        company can significantly improve the quality and coherence of the
        outputs from their hand-written document recognition system.
      - >-
        A Part-of-Speech (PoS) tagger is designed to categorize words in a given
        text into their respective grammatical roles, such as nouns, verbs,
        adjectives, etc. This task is complex due to two primary challenges:
        first, lexical ambiguity arises when a single word can serve multiple
        grammatical functions depending on its context (e.g., "lead" can be a
        noun or a verb), which complicates the tagging process and can result in
        numerous potential interpretations as sentence length increases. Second,
        the presence of out-of-vocabulary words—those not found in the tagger's
        training data—presents a further difficulty, as it necessitates the
        development of strategies for effectively assigning tags to these
        unfamiliar terms. This often requires the implementation of linguistic
        heuristics and context-based inference, which adds an additional layer
        of complexity to the tagging process.
  - source_sentence: >-
      One of your colleagues has recently taken over responsibility for a legacy
      codebase, a library currently used by some of your customers. Before
      making functional changes, your colleague found a bug caused by incorrect
      use of the following method in the codebase:


      public class User {
          /** Indicates whether the user’s browser, if any, has JavaScript enabled. */
          public boolean hasJavascriptEnabled() { … }

          // … other methods, such as getName(), getAge(), ...
      }


      Your colleague believes that this is a bad API. You are reviewing the pull
      request your colleague made to fix this bug. After some discussion and
      additional commits to address feedback, the pull request is ready. You can
      either "squash" the pull request into a single commit, or leave the
      multiple commits as they are. Explain in 1 sentence whether you should
      "squash" and why.
    sentences:
      - >-
        **Answer:** A


        **Plan:**

        1. Establish that the given linear programming problem is a bipartite
        network flow problem and identify the relevant properties of bipartite
        graphs.

        2. Use the property of total unimodularity of the constraint matrix in
        the linear program to show that any basic feasible solution (extreme
        point solution) is integral.

        3. Explain how the degree bounds \(b(v)\) affect the structure of the
        feasible region in the linear program.

        4. Conclude that since all extreme points of the feasible region are
        integral, the solution must also be integral.


        **Steps:**


        1. **Understanding the Problem Context:**
           The linear programming formulation given corresponds to the min-cost perfect \(b\)-matching problem in a bipartite graph. The objective is to minimize the cost of the edges while satisfying the degree constraints for each vertex in the graph, where \(b(v)\) indicates how many edges must be incident to vertex \(v\).

        2. **Total Unimodularity of the Constraint Matrix:**
           The matrix representing the constraints in this linear program is derived from the incidence matrix of the bipartite graph. Each row of the matrix corresponds to a vertex \(v\) and contains coefficients that indicate the edges incident to that vertex. In a bipartite graph, this incidence matrix is totally unimodular.
           - A matrix is said to be totally unimodular if every square submatrix has a determinant of \(0\), \(1\), or \(-1\). This property ensures that every basic feasible solution (which corresponds to an extreme point in the feasible region) of the linear program can be expressed with integer values.

        3. **Impact of Degree Bounds \(b(v)\):**
           Since \(b(v)\) are all non-negative integers, the constraints \( \sum_{e\in E: v \in e} x_e = b(v) \) for all vertices \(v\) ensure that the solution must balance the contributions of edge variables \(x_e\) to meet the exact degree requirements. The total unimodularity of the matrix guarantees that these constraints will yield integer solutions for the variables \(x_e\).

        4. **Conclusion on Integral Solutions:**
           Given that the linear program's constraint matrix is totally unimodular and the constraints are linear combinations of the \(x_e\) variables that must equal the integer degree bounds \(b(v)\), we conclude that all extreme point solutions of this linear program are integral. Thus, if the input graph \(G = (V, E)\) is bipartite, every extreme point solution to the linear programming relaxation is indeed integral.

        After reasoning through the properties of the linear program and the
        implications of total unimodularity in a bipartite context, I confirm
        that the argument is valid and the conclusion is correct.
      - >-
        To differentiate the objective function with respect to \( b_u \), we
        start with the given objective:


        \[

        J = \frac{1}{2} \sum_{(u, m)} \left(f_{um} - r_{um}\right)^2 +
        \frac{\lambda}{2} \left[ \sum_{u \in \mathbf{U}} (b_u^2 +
        \|\mathbf{v}_u\|^2) + \sum_{m \in \mathbf{M}} (b_m^2 +
        \|\mathbf{w}_m\|^2) \right]

        \]


        where \( f_{um} = \langle \mathbf{v}_u, \mathbf{w}_m \rangle + b_u + b_m
        \).


        To find the optimal \( b_u \), we focus only on the terms that involve
        \( b_u \). The relevant part of the objective that includes \( b_u \)
        is:


        \[

        J_u = \frac{1}{2} \sum_{m \in N(u)} \left( \left\langle \mathbf{v}_u,
        \mathbf{w}_m \right\rangle + b_u + b_m - r_{um} \right)^2 +
        \frac{\lambda}{2} b_u^2

        \]


        where \( N(u) \) is the set of movies rated by user \( u \).


        Now, to differentiate \( J_u \) with respect to \( b_u \):


        1. **Differentiate the first term**:
           \[
           \frac{\partial J_u}{\partial b_u} = \sum_{m \in N(u)} \left( \left\langle \mathbf{v}_u, \mathbf{w}_m \right\rangle + b_u + b_m - r_{um} \right)
           \]

        2. **Differentiate the regularization term**:
           \[
           \frac{\partial}{\partial b_u} \left( \frac{\lambda}{2} b_u^2 \right) = \lambda b_u
           \]

        Combining these results, we get:


        \[

        \frac{\partial J_u}{\partial b_u} = \sum_{m \in N(u)} \left(
        \left\langle \mathbf{v}_u, \mathbf{w}_m \right\rangle + b_u + b_m -
        r_{um} \right) + \lambda b_u

        \]


        Setting the derivative to zero to find the optimal \( b_u \):


        \[

        \sum_{m \in N(u)} \left( \left\langle \mathbf{v}_u, \mathbf{w}_m
        \right\rangle + b_u + b_m - r_{um} \right) + \lambda b_u = 0

        \]


        Rearranging gives:


        \[

        \sum_{m \in N(u)} \left\langle \mathbf{v}_u, \mathbf{w}_m \right\rangle
        + |N(u)| b_u + \sum_{m \in N(u)} b_m - \sum_{m \in N(u)} r_{um} +
        \lambda b_u = 0

        \]


        Combine the terms involving \( b_u \):


        \[

        \left( |N(u)| + \lambda \right) b_u = \sum_{m \in N(u)} r_{um} - \sum_{m
        \in N(u)} \left\langle \mathbf{v}_u, \mathbf{w}_m \right\rangle -
        \sum_{m \in N(u)} b_m

        \]


        Now, solving for \( b_u \):


        \[

        b_u = \frac{1}{|N(u)| + \lambda} \left( \sum_{m \in N(u)} r_{um} -
        \sum_{m \in N(u)} \left\langle \mathbf{v}_u, \mathbf{w}_m \right\rangle
        - \sum_{m \in N(u)} b_m \right)

        \]


        This is the optimal value of \( b_u \) when all other parameters are
        fixed. The important point here is that the regularization term
        contributes to the denominator, ensuring that the bias \( b_u \) is
        adjusted appropriately based on the number of ratings \( |N(u)| \) and
        the regularization strength \( \lambda \).
      - >-
        Yes, you should squash the pull request into a single commit to ensure
        that the commit history is clean and focused on the final, complete
        change, avoiding confusion from intermediate commits that may not
        represent valid states of the code.
  - source_sentence: >-
      Suppose we use the Simplex method to solve the following linear program:
      \begin{align*} \textbf{maximize} \hspace{0.8cm} & 2x_1 - x_2 \\
      \textbf{subject to}\hspace{0.8cm} & x_1 - x_2  + s_1  = 1 \\
      \hspace{0.8cm} & \hspace{0.85cm}x_1  + s_2  = 4 \\ \hspace{0.8cm} &
      \hspace{0.85cm} x_2  + s_3 = 2 \\ \hspace{0.8cm} &\hspace{-0.8cm}  x_1,\:
      x_2, \:s_1, \:s_2, \:s_3 \geq 0 \end{align*} At the current step, we have
      the following Simplex tableau: \begin{align*} \hspace{1cm} x_1 &= 1 + x_2
      - s_1 \\ s_2 &= 3 -x_2  + s_1  \\ s_3 &= 2 -x_2  \\ \cline{1-2} z &= 2 +
      x_2 - 2s_1 \end{align*} Write the tableau obtained by executing one
      iteration (pivot) of the Simplex method starting from the above tableau.
    sentences:
      - >-
        To analyze whether we need to introduce constraints on the number of
        Byzantine processes in a non-synchronous environment, we first need to
        understand the properties of Byzantine consistent broadcast (BCB) and
        the implications of Byzantine processes on these properties.


        ### Definitions:

        - Let \( N \) be the total number of processes in the system.

        - Let \( F \) be the maximum number of Byzantine processes.


        ### Properties of Byzantine Consistent Broadcast:

        1. **Validity**: If the designated sender \( S \) is correct, then every
        correct process eventually delivers the message.

        2. **No duplication**: Every correct process delivers at most one
        message.

        3. **Integrity**: If a correct process delivers a message, and \( S \)
        is correct, then \( S \) has previously broadcast the message.

        4. **Consistency**: No two correct processes deliver different messages.


        ### Analyzing Byzantine Processes:

        In a non-synchronous environment, messages may take an unpredictable
        amount of time to be delivered, and processes may operate independently
        without a global clock. This introduces challenges when Byzantine
        processes (which can behave arbitrarily) are present.


        #### Hypothetical Scenarios:

        1. **Scenario with \( N = 3F \)**:
           - Suppose \( N = 3F \), meaning there are exactly three times as many processes as there are Byzantine processes. For example, if \( F = 1 \), then \( N = 3 \). In this case, the processes can be divided into:
             - Correct process A
             - Correct process B
             - Byzantine process C

           In this scenario, if the Byzantine process (C) decides to send a different message than what the correct sender (S) sends, it can mislead the correct processes (A and B). Both A and B may receive different messages if they are not able to distinguish between the correct and Byzantine processes.

        2. **Potential Violation of Properties**:
           - **Validity**: If the correct process A receives a message from the Byzantine process C, it may not be able to determine if the message is valid. This could lead to a situation where A does not deliver the correct message broadcasted by S.
           - **Consistency**: If A and B receive different messages due to the Byzantine behavior of C, this violates the consistency property, as A delivers message \( m_1 \) while B delivers \( m_2 \).

        ### Mathematical Relationships:

        To ensure the properties of BCB hold, particularly the consistency
        property, we must have more correct processes than Byzantine processes.
        The established consensus in distributed systems is that the maximum
        number of Byzantine processes \( F \) can be tolerated if and only if:


        $$ N > 3F $$


        or, more precisely,


        $$ N = 3F + 1 $$


        ### Conclusion:

        Yes, we must introduce constraints on the number of Byzantine processes
        in non-synchronous environments to maintain the integrity and
        consistency properties of Byzantine consistent broadcast. Specifically,
        the condition \( N = 3F + 1 \) must be satisfied to ensure that correct
        processes can always outvote or ignore the Byzantine processes, thereby
        ensuring that the system remains reliable and consistent even in the
        presence of failures or malicious behavior. This constraint is crucial
        for the robustness of the BCB protocol in distributed systems.
      - >-
        To execute one iteration (pivot) of the Simplex method from the given
        tableau, we need to follow these key steps:


        1. **Identify the entering variable:** This is the variable that will
        increase to improve the objective function. In our case, we look at the
        coefficients of the objective function row (z-row). We want to maximize
        z, so we will select the variable with the highest positive coefficient.
        In this tableau, the coefficients for \(x_1\), \(x_2\), and the slack
        variables \(s_1\), \(s_2\), and \(s_3\) in the objective function are
        \(2\), \(1\), \(-2\), \(0\), and \(0\), respectively. The variable
        \(x_1\) has the highest positive coefficient of \(2\), so it will be our
        entering variable.


        2. **Identify the leaving variable:** Next, we need to determine which
        variable will leave the basis. This is done using the minimum ratio
        test. We look at the constraints and find the ratios of the current
        solution values to the coefficients of the entering variable in each
        constraint that has a positive coefficient for that variable.

           From the current tableau:
           - For the first equation, \( x_1 = 1 + x_2 - s_1 \): The coefficient of \(x_1\) is \(1\). If we set \(x_2 = 0\) and \(s_1 = 0\), we can solve for the right-hand side (RHS), which is \(1\). The ratio is \( \frac{1}{1} = 1\).
           - For the second equation, \(s_2 = 3 - x_2 + s_1\): The coefficient of \(x_1\) is \(1\). The RHS is \(3\), so the ratio is again \( \frac{3}{1} = 3\).
           - For the third equation, \(s_3 = 2 - x_2\): The coefficient of \(x_1\) is \(0\), so we can't use this equation.

           The minimum ratio is \(1\) from the first equation, so \(s_1\) will leave the basis.

        3. **Perform the pivot operation:** We pivot on the intersection of the
        entering variable \(x_1\) and the leaving variable \(s_1\). We want to
        express all variables in terms of the new basic variable \(x_1\).

           The tableau before pivoting is:

           \[
           \begin{array}{c|c|c|c|c|c}
           & x_1 & x_2 & s_1 & s_2 & s_3 \\
           \hline
           x_1 & 1 & 1 & -1 & 0 & 0 \\
           s_2 & 0 & 1 & 1 & 1 & 0 \\
           s_3 & 0 & 0 & 0 & 0 & 2 \\
           \hline
           z & 0 & 1 & 2 & 0 & 0 \\
           \end{array}
           \]

           Now we perform row operations to update the tableau:

           - **Row for \(x_1\)**: This will remain unchanged as it becomes our new basic variable.
           - **Row for \(s_2\)**: We will subtract the \(0\) multiplied by row \(x_1\).
           - **Row for \(s_3\)**: This row will also not change as \(x_1\) does not appear.
           - **Row for \(z\)**: We need to replace the \(z\) row to reflect the new basis. The new coefficient of \(z\) will be \(2\) because we have removed \(s_1\) from the basis.

        After performing these operations, we can represent the new tableau:


        \[

        \begin{array}{c|c|c|c|c|c}
           & x_1 & x_2 & s_2 & s_3 \\
           \hline
           x_1 & 1 & 1 & -1 & 0 \\
           s_2 & 0 & 1 & 1 & 1 \\
           s_3 & 0 & 0 & 0 & 2 \\
           \hline
           z & 0 & 1 & 2 & 0 \\
        \end{array}

        \]


        Here, we have successfully updated the tableau after one pivot step of
        the Simplex method. The new solution reflects a state where \(x_1\) is
        now a basic variable. We will repeat this process until we find the
        optimal solution.
      - >-
        Certainly! Let's reevaluate the proof that \( w(S_\ell) = \max_{T \in
        \mathcal{I}: |T| = \ell} w(T) \) for all \( \ell = 1, \ldots, k \) in a
        more rigorous manner.


        ### Overview of the Proof Structure


        We need to show that for each \( \ell \), the greedy algorithm produces
        a set \( S_\ell \) of elements with the maximum weight among all
        independent sets of size \( \ell \) in a matroid \( \mathcal{M} = (E,
        \mathcal{I}) \) with a weight function \( w: E \rightarrow \mathbb{R}
        \).


        ### Step 1: Base Case


        For \( \ell = 1 \):

        - \( S_1 = \{s_1\} \) is the element with the highest weight.

        - The maximum weight of any independent set of size 1 is indeed \(
        \max_{e \in E} w(e) \), and since \( s_1 \) is chosen to be the element
        with the maximum weight, we have:


        \[

        w(S_1) = w(s_1) = \max_{T \in \mathcal{I}: |T| = 1} w(T)

        \]


        This base case holds true.


        ### Step 2: Induction Hypothesis


        Now we assume that for some \( \ell \) where \( 1 \leq \ell < k \):


        \[

        w(S_\ell) = \max_{T \in \mathcal{I}: |T| = \ell} w(T)

        \]


        This means that the greedy choice up to \( \ell \) produces a maximum
        weight independent set of size \( \ell \).


        ### Step 3: Induction Step


        We need to prove that:


        \[

        w(S_{\ell + 1}) = \max_{T \in \mathcal{I}: |T| = \ell + 1} w(T)

        \]


        Let \( S_{\ell + 1} = \{s_1, s_2, \ldots, s_{\ell + 1}\} \).


        #### Constructing the Set \( S_{\ell + 1} \)


        To show this equality, we will analyze any independent set \( T \) of
        size \( \ell + 1 \).


        1. **Case 1: \( s_{\ell + 1} \notin T \)**

           If \( s_{\ell + 1} \) is not in \( T \), then \( T \) must consist of elements among \( \{s_1, s_2, \ldots, s_\ell\} \) and possibly other elements from \( E \). Since \( S_\ell \) consists of the elements with the highest weights, we can state:

           \[
           w(T) \leq w(S_\ell)
           \]

           By the induction hypothesis, this means:

           \[
           w(T) \leq \max_{T' \in \mathcal{I}: |T'| = \ell} w(T') = w(S_\ell)
           \]

        2. **Case 2: \( s_{\ell + 1} \in T \)**

           In this case, we can remove \( s_{\ell + 1} \) from \( T \) to obtain an independent set \( T' \) of size \( \ell \). Thus, we have:

           \[
           w(T) = w(T') + w(s_{\ell + 1})
           \]

           Since \( T' \) is an independent set, we can apply the induction hypothesis, which gives us:

           \[
           w(T') \leq w(S_\ell)
           \]

           Therefore:

           \[
           w(T) = w(T') + w(s_{\ell + 1}) \leq w(S_\ell) + w(s_{\ell + 1})
           \]

        ### Conclusion from the Induction Step


        To compare \( w(S_{\ell + 1}) \) with \( w(T) \), we note:


        - \( S_{\ell + 1} \) contains the highest weight elements, and thus:


        \[

        w(S_{\ell + 1}) = w(S_\ell) + w(s_{\ell + 1})

        \]


        Since \( s_{\ell + 1} \) is the highest weight among the remaining
        elements, we establish the following:


        - If \( s_{\ell + 1} \) is included in \( T \), we have \( w(T) \leq
        w(S_{\ell + 1}) \).

        - If \( s_{\ell + 1} \) is not included, \( w(T) \leq w(S_\ell) \) which
        is already less than or equal to \( w(S_{\ell + 1}) \).


        Thus, we conclude that:


        \[

        w(S_{\ell + 1}) = \max_{T \in \mathcal{I}: |T| = \ell + 1} w(T)

        \]


        ### Final Remarks


        The proof holds without any assumptions being violated. Each step has
        been justified based on the properties of the matroid and the greedy
        choice made at each step, ensuring that the maximum weight independent
        set is correctly identified for all sizes from 1 to \( k \).


        This completes the proof with careful consideration of all cases and
        steps involved. Thank you for your patience!
pipeline_tag: sentence-similarity
library_name: sentence-transformers
metrics:
  - cosine_accuracy@1
  - cosine_accuracy@3
  - cosine_accuracy@5
  - cosine_accuracy@10
  - cosine_precision@1
  - cosine_precision@3
  - cosine_precision@5
  - cosine_precision@10
  - cosine_recall@1
  - cosine_recall@3
  - cosine_recall@5
  - cosine_recall@10
  - cosine_ndcg@10
  - cosine_mrr@10
  - cosine_map@100
model-index:
  - name: BGE base Financial Matryoshka
    results:
      - task:
          type: information-retrieval
          name: Information Retrieval
        dataset:
          name: dim 768
          type: dim_768
        metrics:
          - type: cosine_accuracy@1
            value: 0.2771929824561403
            name: Cosine Accuracy@1
          - type: cosine_accuracy@3
            value: 0.8807017543859649
            name: Cosine Accuracy@3
          - type: cosine_accuracy@5
            value: 0.9298245614035088
            name: Cosine Accuracy@5
          - type: cosine_accuracy@10
            value: 0.9824561403508771
            name: Cosine Accuracy@10
          - type: cosine_precision@1
            value: 0.2771929824561403
            name: Cosine Precision@1
          - type: cosine_precision@3
            value: 0.29356725146198825
            name: Cosine Precision@3
          - type: cosine_precision@5
            value: 0.1859649122807017
            name: Cosine Precision@5
          - type: cosine_precision@10
            value: 0.09824561403508769
            name: Cosine Precision@10
          - type: cosine_recall@1
            value: 0.2771929824561403
            name: Cosine Recall@1
          - type: cosine_recall@3
            value: 0.8807017543859649
            name: Cosine Recall@3
          - type: cosine_recall@5
            value: 0.9298245614035088
            name: Cosine Recall@5
          - type: cosine_recall@10
            value: 0.9824561403508771
            name: Cosine Recall@10
          - type: cosine_ndcg@10
            value: 0.6591255563949661
            name: Cosine Ndcg@10
          - type: cosine_mrr@10
            value: 0.551246170983013
            name: Cosine Mrr@10
          - type: cosine_map@100
            value: 0.552368028142022
            name: Cosine Map@100
      - task:
          type: information-retrieval
          name: Information Retrieval
        dataset:
          name: dim 512
          type: dim_512
        metrics:
          - type: cosine_accuracy@1
            value: 0.2807017543859649
            name: Cosine Accuracy@1
          - type: cosine_accuracy@3
            value: 0.8771929824561403
            name: Cosine Accuracy@3
          - type: cosine_accuracy@5
            value: 0.9298245614035088
            name: Cosine Accuracy@5
          - type: cosine_accuracy@10
            value: 0.9824561403508771
            name: Cosine Accuracy@10
          - type: cosine_precision@1
            value: 0.2807017543859649
            name: Cosine Precision@1
          - type: cosine_precision@3
            value: 0.2923976608187134
            name: Cosine Precision@3
          - type: cosine_precision@5
            value: 0.1859649122807017
            name: Cosine Precision@5
          - type: cosine_precision@10
            value: 0.09824561403508769
            name: Cosine Precision@10
          - type: cosine_recall@1
            value: 0.2807017543859649
            name: Cosine Recall@1
          - type: cosine_recall@3
            value: 0.8771929824561403
            name: Cosine Recall@3
          - type: cosine_recall@5
            value: 0.9298245614035088
            name: Cosine Recall@5
          - type: cosine_recall@10
            value: 0.9824561403508771
            name: Cosine Recall@10
          - type: cosine_ndcg@10
            value: 0.6589926958168313
            name: Cosine Ndcg@10
          - type: cosine_mrr@10
            value: 0.5511487050960734
            name: Cosine Mrr@10
          - type: cosine_map@100
            value: 0.5522544998860789
            name: Cosine Map@100
      - task:
          type: information-retrieval
          name: Information Retrieval
        dataset:
          name: dim 256
          type: dim_256
        metrics:
          - type: cosine_accuracy@1
            value: 0.2771929824561403
            name: Cosine Accuracy@1
          - type: cosine_accuracy@3
            value: 0.8736842105263158
            name: Cosine Accuracy@3
          - type: cosine_accuracy@5
            value: 0.9368421052631579
            name: Cosine Accuracy@5
          - type: cosine_accuracy@10
            value: 0.9894736842105263
            name: Cosine Accuracy@10
          - type: cosine_precision@1
            value: 0.2771929824561403
            name: Cosine Precision@1
          - type: cosine_precision@3
            value: 0.2912280701754386
            name: Cosine Precision@3
          - type: cosine_precision@5
            value: 0.18736842105263155
            name: Cosine Precision@5
          - type: cosine_precision@10
            value: 0.09894736842105262
            name: Cosine Precision@10
          - type: cosine_recall@1
            value: 0.2771929824561403
            name: Cosine Recall@1
          - type: cosine_recall@3
            value: 0.8736842105263158
            name: Cosine Recall@3
          - type: cosine_recall@5
            value: 0.9368421052631579
            name: Cosine Recall@5
          - type: cosine_recall@10
            value: 0.9894736842105263
            name: Cosine Recall@10
          - type: cosine_ndcg@10
            value: 0.6595162220662993
            name: Cosine Ndcg@10
          - type: cosine_mrr@10
            value: 0.5497257031467556
            name: Cosine Mrr@10
          - type: cosine_map@100
            value: 0.5503770544576845
            name: Cosine Map@100
      - task:
          type: information-retrieval
          name: Information Retrieval
        dataset:
          name: dim 128
          type: dim_128
        metrics:
          - type: cosine_accuracy@1
            value: 0.29473684210526313
            name: Cosine Accuracy@1
          - type: cosine_accuracy@3
            value: 0.8701754385964913
            name: Cosine Accuracy@3
          - type: cosine_accuracy@5
            value: 0.9333333333333333
            name: Cosine Accuracy@5
          - type: cosine_accuracy@10
            value: 0.9859649122807017
            name: Cosine Accuracy@10
          - type: cosine_precision@1
            value: 0.29473684210526313
            name: Cosine Precision@1
          - type: cosine_precision@3
            value: 0.2900584795321637
            name: Cosine Precision@3
          - type: cosine_precision@5
            value: 0.18666666666666662
            name: Cosine Precision@5
          - type: cosine_precision@10
            value: 0.09859649122807015
            name: Cosine Precision@10
          - type: cosine_recall@1
            value: 0.29473684210526313
            name: Cosine Recall@1
          - type: cosine_recall@3
            value: 0.8701754385964913
            name: Cosine Recall@3
          - type: cosine_recall@5
            value: 0.9333333333333333
            name: Cosine Recall@5
          - type: cosine_recall@10
            value: 0.9859649122807017
            name: Cosine Recall@10
          - type: cosine_ndcg@10
            value: 0.6666560081123373
            name: Cosine Ndcg@10
          - type: cosine_mrr@10
            value: 0.5603745474798105
            name: Cosine Mrr@10
          - type: cosine_map@100
            value: 0.5612992076149971
            name: Cosine Map@100
      - task:
          type: information-retrieval
          name: Information Retrieval
        dataset:
          name: dim 64
          type: dim_64
        metrics:
          - type: cosine_accuracy@1
            value: 0.27017543859649124
            name: Cosine Accuracy@1
          - type: cosine_accuracy@3
            value: 0.8456140350877193
            name: Cosine Accuracy@3
          - type: cosine_accuracy@5
            value: 0.9333333333333333
            name: Cosine Accuracy@5
          - type: cosine_accuracy@10
            value: 0.9789473684210527
            name: Cosine Accuracy@10
          - type: cosine_precision@1
            value: 0.27017543859649124
            name: Cosine Precision@1
          - type: cosine_precision@3
            value: 0.2818713450292398
            name: Cosine Precision@3
          - type: cosine_precision@5
            value: 0.18666666666666662
            name: Cosine Precision@5
          - type: cosine_precision@10
            value: 0.09789473684210524
            name: Cosine Precision@10
          - type: cosine_recall@1
            value: 0.27017543859649124
            name: Cosine Recall@1
          - type: cosine_recall@3
            value: 0.8456140350877193
            name: Cosine Recall@3
          - type: cosine_recall@5
            value: 0.9333333333333333
            name: Cosine Recall@5
          - type: cosine_recall@10
            value: 0.9789473684210527
            name: Cosine Recall@10
          - type: cosine_ndcg@10
            value: 0.6492874099033707
            name: Cosine Ndcg@10
          - type: cosine_mrr@10
            value: 0.5396101364522413
            name: Cosine Mrr@10
          - type: cosine_map@100
            value: 0.5411050628327516
            name: Cosine Map@100

BGE base Financial Matryoshka

This is a sentence-transformers model finetuned from NovaSearch/stella_en_400M_v5 on the json dataset. It maps sentences & paragraphs to a 1024-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: NovaSearch/stella_en_400M_v5
• Maximum Sequence Length: 512 tokens
• Output Dimensionality: 1024 dimensions
• Similarity Function: Cosine Similarity
• Training Dataset:
  • json
• Language: en
• License: apache-2.0

Model Sources

• Documentation: Sentence Transformers Documentation
• Repository: Sentence Transformers on GitHub
• Hugging Face: Sentence Transformers on Hugging Face

Full Model Architecture

SentenceTransformer(
  (0): Transformer({'max_seq_length': 512, 'do_lower_case': False}) with Transformer model: NewModel 
  (1): Pooling({'word_embedding_dimension': 1024, '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): Dense({'in_features': 1024, 'out_features': 1024, 'bias': True, 'activation_function': 'torch.nn.modules.linear.Identity'})
)

Usage

Direct Usage (Sentence Transformers)

First install the Sentence Transformers library:

pip install -U sentence-transformers

Then you can load this model and run inference.

from sentence_transformers import SentenceTransformer

# Download from the 🤗 Hub
model = SentenceTransformer("cristiano-sartori/stella_finetuned")
# Run inference
sentences = [
    'Suppose we use the Simplex method to solve the following linear program: \\begin{align*} \\textbf{maximize} \\hspace{0.8cm} & 2x_1 - x_2 \\\\ \\textbf{subject to}\\hspace{0.8cm} & x_1 - x_2  + s_1  = 1 \\\\ \\hspace{0.8cm} & \\hspace{0.85cm}x_1  + s_2  = 4 \\\\ \\hspace{0.8cm} & \\hspace{0.85cm} x_2  + s_3 = 2 \\\\ \\hspace{0.8cm} &\\hspace{-0.8cm}  x_1,\\: x_2, \\:s_1, \\:s_2, \\:s_3 \\geq 0 \\end{align*} At the current step, we have the following Simplex tableau: \\begin{align*} \\hspace{1cm} x_1 &= 1 + x_2 - s_1 \\\\ s_2 &= 3 -x_2  + s_1  \\\\ s_3 &= 2 -x_2  \\\\ \\cline{1-2} z &= 2 + x_2 - 2s_1 \\end{align*} Write the tableau obtained by executing one iteration (pivot) of the Simplex method starting from the above tableau.',
    "To execute one iteration (pivot) of the Simplex method from the given tableau, we need to follow these key steps:\n\n1. **Identify the entering variable:** This is the variable that will increase to improve the objective function. In our case, we look at the coefficients of the objective function row (z-row). We want to maximize z, so we will select the variable with the highest positive coefficient. In this tableau, the coefficients for $x_{1}x\_1$, $x_{2}x\_2$, and the slack variables $s_{1}s\_1$, $s_{2}s\_2$, and $s_{3}s\_3$ in the objective function are $22$, $11$, $-2-2$, $00$, and $00$, respectively. The variable $x_{1}x\_1$ has the highest positive coefficient of $22$, so it will be our entering variable.\n\n2. **Identify the leaving variable:** Next, we need to determine which variable will leave the basis. This is done using the minimum ratio test. We look at the constraints and find the ratios of the current solution values to the coefficients of the entering variable in each constraint that has a positive coefficient for that variable.\n\n   From the current tableau:\n   - For the first equation, $x_1=1+x_2-s_{1}x\_1\; =\; 1\; +\; x\_2\; -\; s\_1$: The coefficient of $x_{1}x\_1$ is $11$. If we set $x_2=0x\_2\; =\; 0$ and $s_1=0s\_1\; =\; 0$, we can solve for the right-hand side (RHS), which is $11$. The ratio is $frac11=1\backslash \backslash frac\{1\}\{1\}\; =\; 1$.\n   - For the second equation, $s_2=3-x_2+s_{1}s\_2\; =\; 3\; -\; x\_2\; +\; s\_1$: The coefficient of $x_{1}x\_1$ is $11$. The RHS is $33$, so the ratio is again $frac31=3\backslash \backslash frac\{3\}\{1\}\; =\; 3$.\n   - For the third equation, $s_3=2-x_{2}s\_3\; =\; 2\; -\; x\_2$: The coefficient of $x_{1}x\_1$ is $00$, so we can't use this equation.\n\n   The minimum ratio is $11$ from the first equation, so $s_{1}s\_1$ will leave the basis.\n\n3. **Perform the pivot operation:** We pivot on the intersection of the entering variable $x_{1}x\_1$ and the leaving variable $s_{1}s\_1$. We want to express all variables in terms of the new basic variable $x_{1}x\_1$.\n\n   The tableau before pivoting is:\n\n   \\[\n   \\begin{array}{c|c|c|c|c|c}\n   & x_1 & x_2 & s_1 & s_2 & s_3 \\\\\n   \\hline\n   x_1 & 1 & 1 & -1 & 0 & 0 \\\\\n   s_2 & 0 & 1 & 1 & 1 & 0 \\\\\n   s_3 & 0 & 0 & 0 & 0 & 2 \\\\\n   \\hline\n   z & 0 & 1 & 2 & 0 & 0 \\\\\n   \\end{array}\n   \\]\n\n   Now we perform row operations to update the tableau:\n\n   - **Row for $x_{1}x\_1$**: This will remain unchanged as it becomes our new basic variable.\n   - **Row for $s_{2}s\_2$**: We will subtract the $00$ multiplied by row $x_{1}x\_1$.\n   - **Row for $s_{3}s\_3$**: This row will also not change as $x_{1}x\_1$ does not appear.\n   - **Row for $zz$**: We need to replace the $zz$ row to reflect the new basis. The new coefficient of $zz$ will be $22$ because we have removed $s_{1}s\_1$ from the basis.\n\nAfter performing these operations, we can represent the new tableau:\n\n\\[\n\\begin{array}{c|c|c|c|c|c}\n   & x_1 & x_2 & s_2 & s_3 \\\\\n   \\hline\n   x_1 & 1 & 1 & -1 & 0 \\\\\n   s_2 & 0 & 1 & 1 & 1 \\\\\n   s_3 & 0 & 0 & 0 & 2 \\\\\n   \\hline\n   z & 0 & 1 & 2 & 0 \\\\\n\\end{array}\n\\]\n\nHere, we have successfully updated the tableau after one pivot step of the Simplex method. The new solution reflects a state where $x_{1}x\_1$ is now a basic variable. We will repeat this process until we find the optimal solution.",
    "Certainly! Let's reevaluate the proof that  w(S_\\ell) = \\max_{T \\in \\mathcal{I}: |T| = \\ell} w(T)  for all $ell=1,ldots,k\backslash \backslash ell\; =\; 1,\; \backslash \backslash ldots,\; k$ in a more rigorous manner.\n\n### Overview of the Proof Structure\n\nWe need to show that for each $ell\backslash \backslash ell$, the greedy algorithm produces a set  S_\\ell  of elements with the maximum weight among all independent sets of size $ell\backslash \backslash ell$ in a matroid $mathcalM=(E,mathcalI)\backslash \backslash mathcal\{M\}\; =\; (E,\; \backslash \backslash mathcal\{I\})$ with a weight function $w:ErightarrowmathbbRw:\; E\; \backslash \backslash rightarrow\; \backslash \backslash mathbb\{R\}$.\n\n### Step 1: Base Case\n\nFor $ell=1\backslash \backslash ell\; =\; 1$:\n- $S_1=s_{1}S\_1\; =\; \backslash \backslash \{s\_1\backslash \backslash \}$ is the element with the highest weight.\n- The maximum weight of any independent set of size 1 is indeed $ma{x}_{einE}w(e)\backslash \backslash max\_\{e\; \backslash \backslash in\; E\}\; w(e)$, and since $s_{1}s\_1$ is chosen to be the element with the maximum weight, we have:\n\n\\[\nw(S_1) = w(s_1) = \\max_{T \\in \\mathcal{I}: |T| = 1} w(T)\n\\]\n\nThis base case holds true.\n\n### Step 2: Induction Hypothesis\n\nNow we assume that for some $ell\backslash \backslash ell$ where :\n\n\\[\nw(S_\\ell) = \\max_{T \\in \\mathcal{I}: |T| = \\ell} w(T)\n\\]\n\nThis means that the greedy choice up to $ell\backslash \backslash ell$ produces a maximum weight independent set of size $ell\backslash \backslash ell$.\n\n### Step 3: Induction Step\n\nWe need to prove that:\n\n\\[\nw(S_{\\ell + 1}) = \\max_{T \\in \\mathcal{I}: |T| = \\ell + 1} w(T)\n\\]\n\nLet $S_{ell+1}=s_1,s_2,ldots,s_{ell+1}S\_\{\backslash \backslash ell\; +\; 1\}\; =\; \backslash \backslash \{s\_1,\; s\_2,\; \backslash \backslash ldots,\; s\_\{\backslash \backslash ell\; +\; 1\}\backslash \backslash \}$.\n\n#### Constructing the Set $S_{ell+1}S\_\{\backslash \backslash ell\; +\; 1\}$\n\nTo show this equality, we will analyze any independent set $TT$ of size $ell+1\backslash \backslash ell\; +\; 1$.\n\n1. **Case 1: $s_{ell+1}notinTs\_\{\backslash \backslash ell\; +\; 1\}\; \backslash \backslash notin\; T$**\n\n   If $s_{ell+1}s\_\{\backslash \backslash ell\; +\; 1\}$ is not in $TT$, then $TT$ must consist of elements among  \\{s_1, s_2, \\ldots, s_\\ell\\}  and possibly other elements from $EE$. Since  S_\\ell  consists of the elements with the highest weights, we can state:\n\n   \\[\n   w(T) \\leq w(S_\\ell)\n   \\]\n\n   By the induction hypothesis, this means:\n\n   \\[\n   w(T) \\leq \\max_{T' \\in \\mathcal{I}: |T'| = \\ell} w(T') = w(S_\\ell)\n   \\]\n\n2. **Case 2: $s_{ell+1}inTs\_\{\backslash \backslash ell\; +\; 1\}\; \backslash \backslash in\; T$**\n\n   In this case, we can remove $s_{ell+1}s\_\{\backslash \backslash ell\; +\; 1\}$ from $TT$ to obtain an independent set $T^{\mathrm{\prime }}T\text{'}$ of size $ell\backslash \backslash ell$. Thus, we have:\n\n   \\[\n   w(T) = w(T') + w(s_{\\ell + 1})\n   \\]\n\n   Since $T^{\mathrm{\prime }}T\text{'}$ is an independent set, we can apply the induction hypothesis, which gives us:\n\n   \\[\n   w(T') \\leq w(S_\\ell)\n   \\]\n\n   Therefore:\n\n   \\[\n   w(T) = w(T') + w(s_{\\ell + 1}) \\leq w(S_\\ell) + w(s_{\\ell + 1})\n   \\]\n\n### Conclusion from the Induction Step\n\nTo compare $w(S_{ell+1})w(S\_\{\backslash \backslash ell\; +\; 1\})$ with $w(T)w(T)$, we note:\n\n- $S_{ell+1}S\_\{\backslash \backslash ell\; +\; 1\}$ contains the highest weight elements, and thus:\n\n\\[\nw(S_{\\ell + 1}) = w(S_\\ell) + w(s_{\\ell + 1})\n\\]\n\nSince $s_{ell+1}s\_\{\backslash \backslash ell\; +\; 1\}$ is the highest weight among the remaining elements, we establish the following:\n\n- If $s_{ell+1}s\_\{\backslash \backslash ell\; +\; 1\}$ is included in $TT$, we have $w(T)leqw(S_{ell+1})w(T)\; \backslash \backslash leq\; w(S\_\{\backslash \backslash ell\; +\; 1\})$.\n- If $s_{ell+1}s\_\{\backslash \backslash ell\; +\; 1\}$ is not included,  w(T) \\leq w(S_\\ell)  which is already less than or equal to $w(S_{ell+1})w(S\_\{\backslash \backslash ell\; +\; 1\})$.\n\nThus, we conclude that:\n\n\\[\nw(S_{\\ell + 1}) = \\max_{T \\in \\mathcal{I}: |T| = \\ell + 1} w(T)\n\\]\n\n### Final Remarks\n\nThe proof holds without any assumptions being violated. Each step has been justified based on the properties of the matroid and the greedy choice made at each step, ensuring that the maximum weight independent set is correctly identified for all sizes from 1 to $kk$.\n\nThis completes the proof with careful consideration of all cases and steps involved. Thank you for your patience!",
]
embeddings = model.encode(sentences)
print(embeddings.shape)
# [3, 1024]

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

Evaluation

Metrics

Information Retrieval

• Dataset: dim_768
• Evaluated with InformationRetrievalEvaluator with these parameters:

  {
      "truncate_dim": 768
  }
  

Metric	Value
cosine_accuracy@1	0.2772
cosine_accuracy@3	0.8807
cosine_accuracy@5	0.9298
cosine_accuracy@10	0.9825
cosine_precision@1	0.2772
cosine_precision@3	0.2936
cosine_precision@5	0.186
cosine_precision@10	0.0982
cosine_recall@1	0.2772
cosine_recall@3	0.8807
cosine_recall@5	0.9298
cosine_recall@10	0.9825
cosine_ndcg@10	0.6591
cosine_mrr@10	0.5512
cosine_map@100	0.5524

Information Retrieval

• Dataset: dim_512
• Evaluated with InformationRetrievalEvaluator with these parameters:

  {
      "truncate_dim": 512
  }
  

Metric	Value
cosine_accuracy@1	0.2807
cosine_accuracy@3	0.8772
cosine_accuracy@5	0.9298
cosine_accuracy@10	0.9825
cosine_precision@1	0.2807
cosine_precision@3	0.2924
cosine_precision@5	0.186
cosine_precision@10	0.0982
cosine_recall@1	0.2807
cosine_recall@3	0.8772
cosine_recall@5	0.9298
cosine_recall@10	0.9825
cosine_ndcg@10	0.659
cosine_mrr@10	0.5511
cosine_map@100	0.5523

Information Retrieval

• Dataset: dim_256
• Evaluated with InformationRetrievalEvaluator with these parameters:

  {
      "truncate_dim": 256
  }
  

Metric	Value
cosine_accuracy@1	0.2772
cosine_accuracy@3	0.8737
cosine_accuracy@5	0.9368
cosine_accuracy@10	0.9895
cosine_precision@1	0.2772
cosine_precision@3	0.2912
cosine_precision@5	0.1874
cosine_precision@10	0.0989
cosine_recall@1	0.2772
cosine_recall@3	0.8737
cosine_recall@5	0.9368
cosine_recall@10	0.9895
cosine_ndcg@10	0.6595
cosine_mrr@10	0.5497
cosine_map@100	0.5504

Information Retrieval

• Dataset: dim_128
• Evaluated with InformationRetrievalEvaluator with these parameters:

  {
      "truncate_dim": 128
  }
  

Metric	Value
cosine_accuracy@1	0.2947
cosine_accuracy@3	0.8702
cosine_accuracy@5	0.9333
cosine_accuracy@10	0.986
cosine_precision@1	0.2947
cosine_precision@3	0.2901
cosine_precision@5	0.1867
cosine_precision@10	0.0986
cosine_recall@1	0.2947
cosine_recall@3	0.8702
cosine_recall@5	0.9333
cosine_recall@10	0.986
cosine_ndcg@10	0.6667
cosine_mrr@10	0.5604
cosine_map@100	0.5613

Information Retrieval

• Dataset: dim_64
• Evaluated with InformationRetrievalEvaluator with these parameters:

  {
      "truncate_dim": 64
  }
  

Metric	Value
cosine_accuracy@1	0.2702
cosine_accuracy@3	0.8456
cosine_accuracy@5	0.9333
cosine_accuracy@10	0.9789
cosine_precision@1	0.2702
cosine_precision@3	0.2819
cosine_precision@5	0.1867
cosine_precision@10	0.0979
cosine_recall@1	0.2702
cosine_recall@3	0.8456
cosine_recall@5	0.9333
cosine_recall@10	0.9789
cosine_ndcg@10	0.6493
cosine_mrr@10	0.5396
cosine_map@100	0.5411

Training Details

Training Dataset

json

• Dataset: json
• Size: 1,140 training samples
• Columns: anchor and positive
• Approximate statistics based on the first 1000 samples:

  	anchor	positive
  type	string	string
  details	min: 5 tokens mean: 169.58 tokens max: 512 tokens	min: 3 tokens mean: 375.1 tokens max: 512 tokens

• Samples:

  anchor	positive
  In the following let $\kappa_{1}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)$ and $\kappa_{2}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)$ be two valid kernels. Show that the following is also valid kernel: $\kappa\left(\mathbf{x}, \mathbf{x}^{\prime}\right)=\kappa_{1}\left(\mathbf{x}, \mathbf{x}^{\prime}\right) \kappa_{2}\left(\mathbf{x}, \mathbf{x}^{\prime}\right)$.	To show that the product of two kernels, (\kappa(\mathbf{x}, \mathbf{x}^{\prime}) = \kappa_1(\mathbf{x}, \mathbf{x}^{\prime}) \kappa_2(\mathbf{x}, \mathbf{x}^{\prime})), is also a valid kernel, we can use the property of positive semi-definiteness. ### Step 1: Understanding Kernels A function (\kappa(\mathbf{x}, \mathbf{x}^{\prime})) is a kernel if it is symmetric (i.e., (\kappa(\mathbf{x}, \mathbf{x}^{\prime}) = \kappa(\mathbf{x}^{\prime}, \mathbf{x}))) and positive semi-definite (PSD). A function is PSD if for any finite set of points ({\mathbf{x}_1, \mathbf{x}2, \ldots, \mathbf{x}n}) and any real coefficients (c_1, c_2, \ldots, c_n), the following holds: [ \sum{i=1}^n \sum{j=1}^n c_i c_j \kappa(\mathbf{x}_i, \mathbf{x}_j} \geq 0. ] ### Step 2: Properties of (\kappa_1) and (\kappa_2) Given that (\kappa_1) and (\kappa_2) are kernels, we know: 1. They are symmetric: (\kappa_1(\mathbf{x}, \mathbf{x}^{\prime}) = \kappa_1(\mathbf{x}^{\prime}, \mathbf{x})) and...
  The first annotator rated {a} reviews as positive and the rest as negative. The second annotator rated {b} reviews as positive and the rest as negative. 80 reviews were rated as positive by both annotators. Compute the quality of the above reference using Cohen's Kappa.Give your answer as a numerical value to three decimal places.	To compute Cohen's Kappa, we need to understand the components required for the calculation. Cohen's Kappa (( \kappa )) is a statistic that measures inter-rater agreement for categorical items. The formula for Cohen's Kappa is: [ \kappa = \frac{P_o - P_e}{1 - P_e} ] Where: - ( P_o ) is the observed agreement (the proportion of times both annotators agreed). - ( P_e ) is the expected agreement by chance. Let's break down the problem step by step: 1. Understanding the Data: - Let ( N ) be the total number of reviews. - The first annotator rated ( a ) reviews as positive and ( N - a ) as negative. - The second annotator rated ( b ) reviews as positive and ( N - b ) as negative. - We know that both annotators agreed on 80 reviews being positive. 2. Observed Agreement ( P_o ): - The total number of reviews is ( N ). - The observed agreement ( P_o ) is the proportion of reviews where both annotators agreed on the rating. - Therefore...
  Consider the (toy) grammar $G$ consisting of the following rules: R1: S --> NP VP R2: NP --> NN R3: NP --> Det NN R4: NN --> N R5: NN --> NN NN R6: NN --> NN PNP R7: PNP --> Prep NP R8: VP --> V R9: VP --> Adv V Precisely define the type of grammar G is corresponding to (for that, consider at least the following aspects: dependency-based vs. constituency-based, position in the Chomsky hierarchy, and CNF). Justify your answer for each of the aspects you will be mentioning.	To analyze the grammar ( G ) you've provided, we need to consider several aspects: whether it is dependency-based or constituency-based, its position in the Chomsky hierarchy, and whether it is in Chomsky Normal Form (CNF). ### 1. Dependency-based vs. Constituency-based Definition: - Dependency-based grammar focuses on the relationships between words in a sentence, where a word is dependent on another word (its head). - Constituency-based grammar organizes sentences into nested structures (constituents) that can be replaced with single units. Analysis: The grammar ( G ) is constituency-based because it uses hierarchical structures (Non-terminal symbols like ( S, NP, VP, NN )) to define the grammatical relationships. The rules show that phrases (like Noun Phrases and Verb Phrases) are made up of smaller constituents. For instance, ( S ) is defined as consisting of a noun phrase ( NP ) and a verb phrase ( VP ). This clearly indicates a constituency str...

• Loss: MatryoshkaLoss with these parameters:

  {
      "loss": "MultipleNegativesRankingLoss",
      "matryoshka_dims": [
          768,
          512,
          256,
          128,
          64
      ],
      "matryoshka_weights": [
          1,
          1,
          1,
          1,
          1
      ],
      "n_dims_per_step": -1
  }
  

Training Hyperparameters

Non-Default Hyperparameters

• eval_strategy: epoch
• per_device_train_batch_size: 2
• per_device_eval_batch_size: 16
• gradient_accumulation_steps: 16
• learning_rate: 2e-05
• num_train_epochs: 4
• lr_scheduler_type: cosine
• warmup_ratio: 0.1
• bf16: True
• tf32: False
• load_best_model_at_end: True
• optim: adamw_torch_fused
• batch_sampler: no_duplicates

All Hyperparameters

Click to expand

• overwrite_output_dir: False
• do_predict: False
• eval_strategy: epoch
• prediction_loss_only: True
• per_device_train_batch_size: 2
• per_device_eval_batch_size: 16
• per_gpu_train_batch_size: None
• per_gpu_eval_batch_size: None
• gradient_accumulation_steps: 16
• eval_accumulation_steps: None
• torch_empty_cache_steps: None
• learning_rate: 2e-05
• weight_decay: 0.0
• adam_beta1: 0.9
• adam_beta2: 0.999
• adam_epsilon: 1e-08
• max_grad_norm: 1.0
• num_train_epochs: 4
• max_steps: -1
• lr_scheduler_type: cosine
• lr_scheduler_kwargs: {}
• warmup_ratio: 0.1
• 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: True
• fp16: False
• fp16_opt_level: O1
• half_precision_backend: auto
• bf16_full_eval: False
• fp16_full_eval: False
• tf32: False
• 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: True
• 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}
• 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_fused
• 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: no_duplicates
• multi_dataset_batch_sampler: proportional

Training Logs

Epoch	Step	Training Loss	dim_768_cosine_ndcg@10	dim_512_cosine_ndcg@10	dim_256_cosine_ndcg@10	dim_128_cosine_ndcg@10	dim_64_cosine_ndcg@10
0.2807	10	0.1249	-	-	-	-	-
0.5614	20	0.8091	-	-	-	-	-
0.8421	30	0.0235	-	-	-	-	-
1.0	36	-	0.6463	0.6516	0.6468	0.6425	0.6363
1.1123	40	0.0293	-	-	-	-	-
1.3930	50	0.0474	-	-	-	-	-
1.6737	60	0.0062	-	-	-	-	-
1.9544	70	0.0022	-	-	-	-	-
2.0	72	-	0.6535	0.6501	0.6488	0.6488	0.6454
2.2246	80	0.9144	-	-	-	-	-
2.5053	90	0.0139	-	-	-	-	-
2.7860	100	0.0019	-	-	-	-	-
3.0	108	-	0.6533	0.6582	0.6523	0.6651	0.6499
3.0561	110	0.3805	-	-	-	-	-
3.3368	120	0.0075	-	-	-	-	-
3.6175	130	0.0035	-	-	-	-	-
3.8982	140	0.0012	-	-	-	-	-
4.0	144	-	0.6591	0.659	0.6595	0.6667	0.6493

• The bold row denotes the saved checkpoint.

Framework Versions

• Python: 3.12.8
• Sentence Transformers: 4.1.0
• Transformers: 4.52.4
• PyTorch: 2.7.0+cu126
• Accelerate: 1.3.0
• Datasets: 3.6.0
• Tokenizers: 0.21.0

Citation

BibTeX

Sentence Transformers

@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",
}

MatryoshkaLoss

@misc{kusupati2024matryoshka,
    title={Matryoshka Representation Learning},
    author={Aditya Kusupati and Gantavya Bhatt and Aniket Rege and Matthew Wallingford and Aditya Sinha and Vivek Ramanujan and William Howard-Snyder and Kaifeng Chen and Sham Kakade and Prateek Jain and Ali Farhadi},
    year={2024},
    eprint={2205.13147},
    archivePrefix={arXiv},
    primaryClass={cs.LG}
}

MultipleNegativesRankingLoss

@misc{henderson2017efficient,
    title={Efficient Natural Language Response Suggestion for Smart Reply},
    author={Matthew Henderson and Rami Al-Rfou and Brian Strope and Yun-hsuan Sung and Laszlo Lukacs and Ruiqi Guo and Sanjiv Kumar and Balint Miklos and Ray Kurzweil},
    year={2017},
    eprint={1705.00652},
    archivePrefix={arXiv},
    primaryClass={cs.CL}
}