MNLP_M2_document_encoder / README.md

Stergios-Konstantinidis

Add new SentenceTransformer model

2695b3b verified 9 months ago

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tags:
  - sentence-transformers
  - sentence-similarity
  - feature-extraction
  - generated_from_trainer
  - dataset_size:21000
  - loss:ContrastiveTensionLoss
base_model: sentence-transformers/all-MiniLM-L6-v2
widget:
  - source_sentence: '        "The lemma follows by invoking Lemma 4.1 and Lemma A.1.\n\u220e",'
    sentences:
      - '        "To better address non-stationarity with changing uncertainty, we introduce Location-Scale Noise Model (LSNM) into DDPMs, which relaxes the traditional Additive Noise Model (ANM) by incorporating a contextually changing variance: \ud835\udc18=f\u2062(\ud835\udc17)+g\u2062(\ud835\udc17)\u2062\u03f5\ud835\udc18\ud835\udc53\ud835\udc17\ud835\udc54\ud835\udc17bold-italic-\u03f5\\mathbf{Y}=f(\\mathbf{X})+\\sqrt{g(\\mathbf{X})}\\boldsymbol{\\epsilon}bold_Y = italic_f ( bold_X ) + square-root start_ARG italic_g ( bold_X ) end_ARG bold_italic_\u03f5, where g\u2062(\ud835\udc17)\ud835\udc54\ud835\udc17g(\\mathbf{X})italic_g ( bold_X ) is an \ud835\udc17\ud835\udc17\\mathbf{X}bold_X-dependent variance model. LSNM is capable of modeling both the contextual mean through f\u2062(\ud835\udc17)\ud835\udc53\ud835\udc17f(\\mathbf{X})italic_f ( bold_X ) and the contextual uncertainty through g\u2062(\ud835\udc17)\ud835\udc54\ud835\udc17\\sqrt{g(\\mathbf{X})}square-root start_ARG italic_g ( bold_X ) end_ARG. In the special case where g\u2062(\ud835\udc17)\u22611\ud835\udc54\ud835\udc171g(\\mathbf{X})\\equiv 1italic_g ( bold_X ) \u2261 1, this simplifies to the standard ANM. Building upon this more flexible and expressive assumption, we propose the Non-stationary Diffusion Model (NsDiff) framework, which provides an uncertainty-aware noise schedule for both forward and reverse diffusion processes. In summary, our contributions are as:\n\n\n\u2022\n\nWe observe that the ANM is inadequate for capturing the varying uncertainty and propose a novel framework that integrates LSNM to allow for explict uncertainty modeling. This work is the first attempt to introduce LSNM into probabilistic time series forecasting.\n\n\n\n\u2022\n\nTo fundamentally elevate the noise modeling capabilities of DDPM, we seamlessly integrate time-varying variances into the core diffusion process through an uncertainty-aware noise schedule that dynamically adapts the noise variance at each step.\n\n\n\n\n\u2022\n\nExperimental results indicate that NsDiff achieves superior performance in capturing uncertainty. Specifically, in comparison to the second-best recent baseline TMDM, NsDiff improves up to 66.3% on real-world datasets and 88.3% on synthetic datasets.",'
      - '        "The deep neural network representation of the Bifrost simulations is highly compressed compared to the original Bifrost data: the deep neural network has 44,261 floating point values whereas the Bifrost simulation cube has 96\u22c596\u22c564\u22c520=11,796,480\u22c5969664201179648096\\cdot 96\\cdot 64\\cdot 20=11,796,48096 \u22c5 96 \u22c5 64 \u22c5 20 = 11 , 796 , 480 floating point values. This corresponds to a compression by a factor of 267; this compression factor may be different for other numerical simulations and depends on their smoothness. In addition, the deep neural network can be evaluated at any point in space and time covered by the simulations, therefore enabling a trivial way to interpolate between grid points; furthermore, gradients are calculate with high efficiency with automatic differentiation. As such, it might be worth considering releasing deep-neural-network representations of (magneto)hydrodynamic simulations.",'
      - '        "\u03f5y\u2062(\u03bc)={1nt\u2062\u2211i=nkntey\u2062(ti,\u03bc)=1nt\u2062\u2211i=nknt|y~\u2062(ti,\u03bc)\u2212y\u2062(ti,\u03bc)|if\u00a0\u20621nt\u2062\u2211i=nknt|y\u2062(ti,\u03bc)|\u22641,1nt\u2062\u2211i=nkntey,r\u2062e\u2062l\u2062(ti,\u03bc)=1nt\u2062\u2211i=nknt|y~\u2062(ti,\u03bc)\u2212y\u2062(ti,\u03bc)|/|y\u2062(ti,\u03bc)|if\u00a0\u20621nt\u2062\u2211i=nknt|y\u2062(ti,\u03bc)|>1.subscriptitalic-\u03f5\ud835\udc66\ud835\udf07cases1subscript\ud835\udc5b\ud835\udc61superscriptsubscript\ud835\udc56subscript\ud835\udc5b\ud835\udc58subscript\ud835\udc5b\ud835\udc61subscript\ud835\udc52\ud835\udc66subscript\ud835\udc61\ud835\udc56\ud835\udf071subscript\ud835\udc5b\ud835\udc61superscriptsubscript\ud835\udc56subscript\ud835\udc5b\ud835\udc58subscript\ud835\udc5b\ud835\udc61~\ud835\udc66subscript\ud835\udc61\ud835\udc56\ud835\udf07\ud835\udc66subscript\ud835\udc61\ud835\udc56\ud835\udf07if\u00a01subscript\ud835\udc5b\ud835\udc61superscriptsubscript\ud835\udc56subscript\ud835\udc5b\ud835\udc58subscript\ud835\udc5b\ud835\udc61\ud835\udc66subscript\ud835\udc61\ud835\udc56\ud835\udf0711subscript\ud835\udc5b\ud835\udc61superscriptsubscript\ud835\udc56subscript\ud835\udc5b\ud835\udc58subscript\ud835\udc5b\ud835\udc61subscript\ud835\udc52\ud835\udc66\ud835\udc5f\ud835\udc52\ud835\udc59subscript\ud835\udc61\ud835\udc56\ud835\udf071subscript\ud835\udc5b\ud835\udc61superscriptsubscript\ud835\udc56subscript\ud835\udc5b\ud835\udc58subscript\ud835\udc5b\ud835\udc61~\ud835\udc66subscript\ud835\udc61\ud835\udc56\ud835\udf07\ud835\udc66subscript\ud835\udc61\ud835\udc56\ud835\udf07\ud835\udc66subscript\ud835\udc61\ud835\udc56\ud835\udf07if\u00a01subscript\ud835\udc5b\ud835\udc61superscriptsubscript\ud835\udc56subscript\ud835\udc5b\ud835\udc58subscript\ud835\udc5b\ud835\udc61\ud835\udc66subscript\ud835\udc61\ud835\udc56\ud835\udf071\\centering\\epsilon_{y}(\\mu)=\\begin{cases}\\frac{1}{n_{t}}\\sum\\limits_{i=n_{k}}^%\n{n_{t}}e_{y}(t_{i},\\mu)=\\frac{1}{n_{t}}\\sum\\limits_{i=n_{k}}^{n_{t}}|\\tilde{y}%\n(t_{i},\\mu)-y(t_{i},\\mu)|&\\text{if }\\frac{1}{n_{t}}\\sum\\limits_{i=n_{k}}^{n_{t%\n}}|y(t_{i},\\mu)|\\leq 1,\\\\\n\\frac{1}{n_{t}}\\sum\\limits_{i=n_{k}}^{n_{t}}e_{y,rel}(t_{i},\\mu)=\\frac{1}{n_{t%\n}}\\sum\\limits_{i=n_{k}}^{n_{t}}|\\tilde{y}(t_{i},\\mu)-y(t_{i},\\mu)|/|y(t_{i},%\n\\mu)|&\\text{if }\\frac{1}{n_{t}}\\sum\\limits_{i=n_{k}}^{n_{t}}|y(t_{i},\\mu)|>1.%\n\\end{cases}\\@add@centeringitalic_\u03f5 start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_\u03bc ) = { start_ROW start_CELL divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG \u2211 start_POSTSUBSCRIPT italic_i = italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) = divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG \u2211 start_POSTSUBSCRIPT italic_i = italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUPERSCRIPT | over~ start_ARG italic_y end_ARG ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) - italic_y ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) | end_CELL start_CELL if divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG \u2211 start_POSTSUBSCRIPT italic_i = italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUPERSCRIPT | italic_y ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) | \u2264 1 , end_CELL end_ROW start_ROW start_CELL divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG \u2211 start_POSTSUBSCRIPT italic_i = italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUPERSCRIPT italic_e start_POSTSUBSCRIPT italic_y , italic_r italic_e italic_l end_POSTSUBSCRIPT ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) = divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG \u2211 start_POSTSUBSCRIPT italic_i = italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUPERSCRIPT | over~ start_ARG italic_y end_ARG ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) - italic_y ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) | / | italic_y ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) | end_CELL start_CELL if divide start_ARG 1 end_ARG start_ARG italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_ARG \u2211 start_POSTSUBSCRIPT italic_i = italic_n start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n start_POSTSUBSCRIPT italic_t end_POSTSUBSCRIPT end_POSTSUPERSCRIPT | italic_y ( italic_t start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_\u03bc ) | > 1 . end_CELL end_ROW\n\n(12)",'
  - source_sentence: '        "While significant research addresses design tolerance optimisation in manufacturing, there is very little focus on production inspection machines such as AOIs for manufactured products. For AOIs inspecting PCBs, each component may demand a distinct tolerance for each type of inspection, leading to thousands of possible scenarios. Consequently, a general paradigm is needed that accommodates inspection of all components, including new introductions that the system has not previously encountered.",'
    sentences:
      - '        "Indeed, for any e\u2208D0\ud835\udc52subscript\ud835\udc370e\\in D_{0}italic_e \u2208 italic_D start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, there exists a \u03b4\ud835\udeff\\deltaitalic_\u03b4-tube Te\u03b4\u2062(ae)subscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udc52subscript\ud835\udc4e\ud835\udc52T^{\\delta}_{e}(a_{e})italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) centred at some ae\u2208Asubscript\ud835\udc4e\ud835\udc52\ud835\udc34a_{e}\\in Aitalic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT \u2208 italic_A such that\n\n\n\n1|Te\u03b4\u2062(ae)|n\u2062|E\u2229Te\u03b4\u2062(ae)|n>\u03bb.1subscriptsubscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udc52subscript\ud835\udc4e\ud835\udc52\ud835\udc5bsubscript\ud835\udc38subscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udc52subscript\ud835\udc4e\ud835\udc52\ud835\udc5b\ud835\udf06\\frac{1}{\\left|T^{\\delta}_{e}(a_{e})\\right|_{n}}\\left|E\\cap T^{\\delta}_{e}(a_{%\ne})\\right|_{n}>\\lambda.divide start_ARG 1 end_ARG start_ARG | italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG | italic_E \u2229 italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT > italic_\u03bb .\n\n\n\nSince Emsubscript\ud835\udc38\ud835\udc5aE_{m}italic_E start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT and E\u00af\u00af\ud835\udc38\\overline{E}over\u00af start_ARG italic_E end_ARG form a partition of E\ud835\udc38Eitalic_E, we obtain\n\n\n\n1|Te\u03b4\u2062(ae)|n\u2062|Em\u2229Te\u03b4\u2062(ae)|n+1|Te\u03b4\u2062(ae)|n\u2062|E\u00af\u2229Te\u03b4\u2062(ae)|n>\u03bb.1subscriptsubscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udc52subscript\ud835\udc4e\ud835\udc52\ud835\udc5bsubscriptsubscript\ud835\udc38\ud835\udc5asubscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udc52subscript\ud835\udc4e\ud835\udc52\ud835\udc5b1subscriptsubscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udc52subscript\ud835\udc4e\ud835\udc52\ud835\udc5bsubscript\u00af\ud835\udc38subscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udc52subscript\ud835\udc4e\ud835\udc52\ud835\udc5b\ud835\udf06\\frac{1}{\\left|T^{\\delta}_{e}(a_{e})\\right|_{n}}\\left|E_{m}\\cap T^{\\delta}_{e}%\n(a_{e})\\right|_{n}+\\frac{1}{\\left|T^{\\delta}_{e}(a_{e})\\right|_{n}}\\left|%\n\\overline{E}\\cap T^{\\delta}_{e}(a_{e})\\right|_{n}>\\lambda.divide start_ARG 1 end_ARG start_ARG | italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG | italic_E start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT \u2229 italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT + divide start_ARG 1 end_ARG start_ARG | italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG | over\u00af start_ARG italic_E end_ARG \u2229 italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_e end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT > italic_\u03bb .\n\n\n\nThus, at least one of the terms on the left-hand side must be greater than \u03bb2\ud835\udf062\\frac{\\lambda}{2}divide start_ARG italic_\u03bb end_ARG start_ARG 2 end_ARG, implying e\u2208Dm\u222aD\u00af\ud835\udc52subscript\ud835\udc37\ud835\udc5a\u00af\ud835\udc37e\\in D_{m}\\cup\\overline{D}italic_e \u2208 italic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT \u222a over\u00af start_ARG italic_D end_ARG from the definition (3.14) and (3.19). Since\n\n\n\n|Dm|n\u22121+|D\u00af|n\u22121\u2a7e|D0|n\u22121=\u03b50subscriptsubscript\ud835\udc37\ud835\udc5a\ud835\udc5b1subscript\u00af\ud835\udc37\ud835\udc5b1subscriptsubscript\ud835\udc370\ud835\udc5b1subscript\ud835\udf000\\left|D_{m}\\right|_{n-1}+\\left|\\overline{D}\\right|_{n-1}\\geqslant\\left|D_{0}%\n\\right|_{n-1}=\\varepsilon_{0}| italic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT + | over\u00af start_ARG italic_D end_ARG | start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT \u2a7e | italic_D start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT = italic_\u03b5 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT\n\n\n\nand the stopping condition ensures\n\n\n\n|Dm|n\u22121<14\u2062\u03b50,subscriptsubscript\ud835\udc37\ud835\udc5a\ud835\udc5b114subscript\ud835\udf000\\left|D_{m}\\right|_{n-1}<\\frac{1}{4}\\varepsilon_{0},| italic_D start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT | start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT < divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_\u03b5 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ,\n\n\n\nit follows that\n\n\n(3.20)\n\n|D\u00af|n\u22121\u2a7e14\u2062\u03b50.subscript\u00af\ud835\udc37\ud835\udc5b114subscript\ud835\udf000\\left|\\overline{D}\\right|_{n-1}\\geqslant\\frac{1}{4}\\varepsilon_{0}.| over\u00af start_ARG italic_D end_ARG | start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT \u2a7e divide start_ARG 1 end_ARG start_ARG 4 end_ARG italic_\u03b5 start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT .\n\n\n\nFor any \u03be\u2208D\u00af\ud835\udf09\u00af\ud835\udc37\\xi\\in\\overline{D}italic_\u03be \u2208 over\u00af start_ARG italic_D end_ARG, there exists a \u03b4\ud835\udeff\\deltaitalic_\u03b4-tube T\u03be\u03b4\u2062(a\u03be)subscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09T^{\\delta}_{\\xi}(a_{\\xi})italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) centred at a\u03be\u2208Asubscript\ud835\udc4e\ud835\udf09\ud835\udc34a_{\\xi}\\in Aitalic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT \u2208 italic_A such that\n\n\n\n1|T\u03be\u03b4\u2062(a\u03be)|n\u2062|\u22c3i=0m\u22121(E\u2229\u212ci)\u2229T\u03be\u03b4\u2062(a\u03be)|n>\u03bb2.1subscriptsubscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09\ud835\udc5bsubscriptsuperscriptsubscript\ud835\udc560\ud835\udc5a1\ud835\udc38subscript\u212c\ud835\udc56subscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09\ud835\udc5b\ud835\udf062\\frac{1}{\\left|T^{\\delta}_{\\xi}(a_{\\xi})\\right|_{n}}\\left|\\bigcup_{i=0}^{m-1}(%\nE\\cap\\mathcal{B}_{i})\\cap T^{\\delta}_{\\xi}(a_{\\xi})\\right|_{n}>\\frac{\\lambda}{%\n2}.divide start_ARG 1 end_ARG start_ARG | italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG | \u22c3 start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m - 1 end_POSTSUPERSCRIPT ( italic_E \u2229 caligraphic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) \u2229 italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT > divide start_ARG italic_\u03bb end_ARG start_ARG 2 end_ARG .\n\n\n\nThis implies\n\n\n(3.21)\n\n\u2211i=0m\u22121|\u212ci\u2229T\u03be\u03b4\u2062(a\u03be)|n|T\u03be\u03b4\u2062(a\u03be)|n\u2a7e\u2211i=0m\u22121|(E\u2229\u212ci)\u2229T\u03be\u03b4\u2062(a\u03be)|n|T\u03be\u03b4\u2062(a\u03be)|n\u2a7e|\u22c3i=0m\u22121(E\u2229\u212ci)\u2229T\u03be\u03b4\u2062(a\u03be)|n|T\u03be\u03b4\u2062(a\u03be)|n>\u03bb2superscriptsubscript\ud835\udc560\ud835\udc5a1subscriptsubscript\u212c\ud835\udc56subscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09\ud835\udc5bsubscriptsubscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09\ud835\udc5bsuperscriptsubscript\ud835\udc560\ud835\udc5a1subscript\ud835\udc38subscript\u212c\ud835\udc56subscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09\ud835\udc5bsubscriptsubscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09\ud835\udc5bsubscriptsuperscriptsubscript\ud835\udc560\ud835\udc5a1\ud835\udc38subscript\u212c\ud835\udc56subscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09\ud835\udc5bsubscriptsubscriptsuperscript\ud835\udc47\ud835\udeff\ud835\udf09subscript\ud835\udc4e\ud835\udf09\ud835\udc5b\ud835\udf062\\begin{split}\\frac{\\sum_{i=0}^{m-1}\\left|\\mathcal{B}_{i}\\cap T^{\\delta}_{\\xi}(%\na_{\\xi})\\right|_{n}}{\\left|T^{\\delta}_{\\xi}(a_{\\xi})\\right|_{n}}&\\geqslant%\n\\frac{\\sum_{i=0}^{m-1}\\left|(E\\cap\\mathcal{B}_{i})\\cap T^{\\delta}_{\\xi}(a_{\\xi%\n})\\right|_{n}}{\\left|T^{\\delta}_{\\xi}(a_{\\xi})\\right|_{n}}\\\\\n&\\geqslant\\frac{\\left|\\bigcup_{i=0}^{m-1}(E\\cap\\mathcal{B}_{i})\\cap T^{\\delta}%\n_{\\xi}(a_{\\xi})\\right|_{n}}{\\left|T^{\\delta}_{\\xi}(a_{\\xi})\\right|_{n}}>\\frac{%\n\\lambda}{2}\\end{split}start_ROW start_CELL divide start_ARG \u2211 start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m - 1 end_POSTSUPERSCRIPT | caligraphic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT \u2229 italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG | italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG end_CELL start_CELL \u2a7e divide start_ARG \u2211 start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m - 1 end_POSTSUPERSCRIPT | ( italic_E \u2229 caligraphic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) \u2229 italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG | italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG end_CELL end_ROW start_ROW start_CELL end_CELL start_CELL \u2a7e divide start_ARG | \u22c3 start_POSTSUBSCRIPT italic_i = 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_m - 1 end_POSTSUPERSCRIPT ( italic_E \u2229 caligraphic_B start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ) \u2229 italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG start_ARG | italic_T start_POSTSUPERSCRIPT italic_\u03b4 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_\u03be end_POSTSUBSCRIPT ) | start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_ARG > divide start_ARG italic_\u03bb end_ARG start_ARG 2 end_ARG end_CELL end_ROW",'
      - '        "In [kipvar], the authors first add and subtract terms to\nexplicitly express\nIn\u2062(f,\u22c5)subscript\ud835\udc3c\ud835\udc5b\ud835\udc53\u22c5I_{n}(f,\\cdot)italic_I start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_f , \u22c5 ) in terms of\nDynkin martingale and then pass to\nthe limit \u03bb\u21930\u2193\ud835\udf060\\lambda\\downarrow 0italic_\u03bb \u2193 0, before\nanalyzing that result in a second limit as\nn\u2192\u221e\u2192\ud835\udc5bn\\to\\inftyitalic_n \u2192 \u221e. This is the approach of\n[varadhan95, liggett99, landim] as well.\nThe essential idea of the present proof is to first note that\nfor f\u2208\ud835\udc9f(\u2212A^)\u221212\u2283\u211bA^\ud835\udc53subscript\ud835\udc9fsuperscript^\ud835\udc3412superset-ofsubscript\u211b^\ud835\udc34f\\in\\mathscr{D}_{(-\\hat{A})^{-\\frac{1}{2}}}\\supset\\mathscr{R}_{\\hat{A}}italic_f \u2208 script_D start_POSTSUBSCRIPT ( - over^ start_ARG italic_A end_ARG ) start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT end_POSTSUBSCRIPT \u2283 script_R start_POSTSUBSCRIPT over^ start_ARG italic_A end_ARG end_POSTSUBSCRIPT, the sequence\n\u039bn\u2062(f,\u03bbn,\u22c5)subscript\u039b\ud835\udc5b\ud835\udc53subscript\ud835\udf06\ud835\udc5b\u22c5\\Lambda_{n}(f,\\lambda_{n},\\cdot)roman_\u039b start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_f , italic_\u03bb start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , \u22c5 ) converges to zero\nin probability as n\u2192\u221e\u2192\ud835\udc5bn\\to\\inftyitalic_n \u2192 \u221e for a choice of the sequence \u03bbnsubscript\ud835\udf06\ud835\udc5b\\lambda_{n}italic_\u03bb start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT\ntending to zero. From this it follows that\nIn\u2062(f,\u22c5)subscript\ud835\udc3c\ud835\udc5b\ud835\udc53\u22c5I_{n}(f,\\cdot)italic_I start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_f , \u22c5 ) and In\u2062(f,\u22c5)\u2212\u039bn\u2062(f,\u03bbn,\u22c5)\u2261An\u2062(f,\u03bbn,\u22c5)subscript\ud835\udc3c\ud835\udc5b\ud835\udc53\u22c5subscript\u039b\ud835\udc5b\ud835\udc53subscript\ud835\udf06\ud835\udc5b\u22c5subscript\ud835\udc34\ud835\udc5b\ud835\udc53subscript\ud835\udf06\ud835\udc5b\u22c5I_{n}(f,\\cdot)-\\Lambda_{n}(f,\\lambda_{n},\\cdot)\\equiv A_{n}(f,\\lambda_{n},\\cdot)italic_I start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_f , \u22c5 ) - roman_\u039b start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_f , italic_\u03bb start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , \u22c5 ) \u2261 italic_A start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_f , italic_\u03bb start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT , \u22c5 ) have the same limit distribution, provided that the limit exists.\nThe proof is then completed by showing that the latter\nlimit exists and can be obtained by an\nargument using Theorem 1 in which n\ud835\udc5bnitalic_n tends to infinity for a\nfixed small, but positive\n\u03bb\u2113subscript\ud835\udf06\u2113\\lambda_{\\ell}italic_\u03bb start_POSTSUBSCRIPT roman_\u2113 end_POSTSUBSCRIPT, to be determined. Thus, this new\nproof exhibits the asymptotic distribution of\n1n\u2062\u222b0n\u2062tf\u2062(X\u2062(s))\u2062\ud835\udc51s,t\u226501\ud835\udc5bsuperscriptsubscript0\ud835\udc5b\ud835\udc61\ud835\udc53\ud835\udc4b\ud835\udc60differential-d\ud835\udc60\ud835\udc610\\frac{1}{\\sqrt{n}}\\int_{0}^{nt}f(X(s))ds,t\\geq 0divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_n end_ARG end_ARG \u222b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n italic_t end_POSTSUPERSCRIPT italic_f ( italic_X ( italic_s ) ) italic_d italic_s , italic_t \u2265 0,\nf\u2208\ud835\udc9f(\u2212A^)\u221212\ud835\udc53subscript\ud835\udc9fsuperscript^\ud835\udc3412f\\in\\mathscr{D}_{(-\\hat{A})^{-\\frac{1}{2}}}italic_f \u2208 script_D start_POSTSUBSCRIPT ( - over^ start_ARG italic_A end_ARG ) start_POSTSUPERSCRIPT - divide start_ARG 1 end_ARG start_ARG 2 end_ARG end_POSTSUPERSCRIPT end_POSTSUBSCRIPT, explicitly\nas the limit of\n1n\u2062\u222b0n\u2062tA^\u2062R\u03bbn\u2062f\u2062(X\u2062(s)),t\u226501\ud835\udc5bsuperscriptsubscript0\ud835\udc5b\ud835\udc61^\ud835\udc34subscript\ud835\udc45subscript\ud835\udf06\ud835\udc5b\ud835\udc53\ud835\udc4b\ud835\udc60\ud835\udc610\\frac{1}{\\sqrt{n}}\\int_{0}^{nt}\\hat{A}R_{\\lambda_{n}}f(X(s)),t\\geq 0divide start_ARG 1 end_ARG start_ARG square-root start_ARG italic_n end_ARG end_ARG \u222b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_n italic_t end_POSTSUPERSCRIPT over^ start_ARG italic_A end_ARG italic_R start_POSTSUBSCRIPT italic_\u03bb start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_f ( italic_X ( italic_s ) ) , italic_t \u2265 0,\nA^\u2062R\u03bbn\u2062f\u2208\u211bA^^\ud835\udc34subscript\ud835\udc45subscript\ud835\udf06\ud835\udc5b\ud835\udc53subscript\u211b^\ud835\udc34\\hat{A}R_{\\lambda_{n}}f\\in\\mathscr{R}_{\\hat{A}}over^ start_ARG italic_A end_ARG italic_R start_POSTSUBSCRIPT italic_\u03bb start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT end_POSTSUBSCRIPT italic_f \u2208 script_R start_POSTSUBSCRIPT over^ start_ARG italic_A end_ARG end_POSTSUBSCRIPT,\nfor a sequence of positive \u201ctuning\u201dparameters \u03bbnsubscript\ud835\udf06\ud835\udc5b\\lambda_{n}italic_\u03bb start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT.\nSo this new approach\nmay have added value in\ncomputational and further theoretical refinements of the fclt.",'
      - '        "Few-shot Voice Cloning: This follows the central concept of speaker adaptation. However, the difference is the amount of data required. Thus, the reference audio can range from a few seconds to a maximum of 5 minutes, which is decided based on previous work, and anything more is challenging to obtain in real-life scenarios.",'
  - source_sentence: '        "For any \u03b3\u2208(0,2\u2062d)\ud835\udefe02\ud835\udc51\\gamma\\in(0,\\sqrt{2d})italic_\u03b3 \u2208 ( 0 , square-root start_ARG 2 italic_d end_ARG ), define a stochastic process\n{P\u03b3(\u03bb)\u2062(\ud835\udc2d):\ud835\udc2d\u2208[0,1]d}conditional-setsuperscriptsubscript\ud835\udc43\ud835\udefe\ud835\udf06\ud835\udc2d\ud835\udc2dsuperscript01\ud835\udc51\\{P_{\\gamma}^{(\\lambda)}(\\mathbf{t}):\\mathbf{t}\\in[0,1]^{d}\\}{ italic_P start_POSTSUBSCRIPT italic_\u03b3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_\u03bb ) end_POSTSUPERSCRIPT ( bold_t ) : bold_t \u2208 [ 0 , 1 ] start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT } by\n\n\n(3.52)\n\nP\u03b3(\u03bb)\u2062(\ud835\udc2d):=exp\u2061(\u03b3\u2062Z\u03bb\u2062(\ud835\udc2d)\u2212\u03b322\u2062\ud835\udd3c\u2062[Z\u03bb\u2062(\ud835\udc2d)2])=exp\u2061(\u03b3\u2062Z\u03bb\u2062(\ud835\udc2d)\u2212\u03b322\u2062R\u03bb\u2062(\ud835\udc2d,\ud835\udc2d)).assignsuperscriptsubscript\ud835\udc43\ud835\udefe\ud835\udf06\ud835\udc2d\ud835\udefesubscript\ud835\udc4d\ud835\udf06\ud835\udc2dsuperscript\ud835\udefe22\ud835\udd3cdelimited-[]subscript\ud835\udc4d\ud835\udf06superscript\ud835\udc2d2\ud835\udefesubscript\ud835\udc4d\ud835\udf06\ud835\udc2dsuperscript\ud835\udefe22subscript\ud835\udc45\ud835\udf06\ud835\udc2d\ud835\udc2d\\displaystyle P_{\\gamma}^{(\\lambda)}(\\mathbf{t}):=\\exp\\Big{(}\\gamma Z_{\\lambda%\n}(\\mathbf{t})-\\frac{\\gamma^{2}}{2}\\mathbb{E}[Z_{\\lambda}(\\mathbf{t})^{2}]\\Big{%\n)}=\\exp\\Big{(}\\gamma Z_{\\lambda}(\\mathbf{t})-\\frac{\\gamma^{2}}{2}R_{\\lambda}(%\n\\mathbf{t},\\mathbf{t})\\Big{)}.italic_P start_POSTSUBSCRIPT italic_\u03b3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_\u03bb ) end_POSTSUPERSCRIPT ( bold_t ) := roman_exp ( italic_\u03b3 italic_Z start_POSTSUBSCRIPT italic_\u03bb end_POSTSUBSCRIPT ( bold_t ) - divide start_ARG italic_\u03b3 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG blackboard_E [ italic_Z start_POSTSUBSCRIPT italic_\u03bb end_POSTSUBSCRIPT ( bold_t ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] ) = roman_exp ( italic_\u03b3 italic_Z start_POSTSUBSCRIPT italic_\u03bb end_POSTSUBSCRIPT ( bold_t ) - divide start_ARG italic_\u03b3 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG italic_R start_POSTSUBSCRIPT italic_\u03bb end_POSTSUBSCRIPT ( bold_t , bold_t ) ) .",'
    sentences:
      - '        "In this section, we highlight open challenges and future directions in network-level ISAC design and the practical implementation of distributed ISAC systems.",'
      - '}'
      - '        "Warning: As before, we need to restrict ourselves to a smaller class of perturbation data (i.e. sufficiently small Hamiltonian perturbations) to ensure that the element on the right is in \u039b0subscript\u039b0\\Lambda_{0}roman_\u039b start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT, in other words such that for any quilted strip u\u00af\u00af\ud835\udc62\\underline{u}under\u00af start_ARG italic_u end_ARG we have \u03c9\u2062(u\u00af)=0\ud835\udf14\u00af\ud835\udc620\\omega(\\underline{u})=0italic_\u03c9 ( under\u00af start_ARG italic_u end_ARG ) = 0 if and only if [u\u00af]=0delimited-[]\u00af\ud835\udc620[\\underline{u}]=0[ under\u00af start_ARG italic_u end_ARG ] = 0.",'
  - source_sentence: '        "For the regular planar lattice graphs, \ud835\udca2\u25b3,\ud835\udca2\u25a1,\ud835\udca2\u2394subscript\ud835\udca2\u25b3subscript\ud835\udca2\u25a1subscript\ud835\udca2\u2394\\mathcal{G}_{\\triangle},\\,\\mathcal{G}_{\\square},\\,\\mathcal{G}_{\\hexagon}caligraphic_G start_POSTSUBSCRIPT \u25b3 end_POSTSUBSCRIPT , caligraphic_G start_POSTSUBSCRIPT \u25a1 end_POSTSUBSCRIPT , caligraphic_G start_POSTSUBSCRIPT \u2394 end_POSTSUBSCRIPT,\n\n\n\nvol\u27c2\u2062(G)=vol\u2062(G)=vol\u25c6\u2062(G)+vol\u25c6\u2062(G\u2217)=2\u2062\u03c0\u2062m\u2062(p\u2062(z,w))=2\u2062\u03c0\u2062zGfd.superscriptvolperpendicular-to\ud835\udc3avol\ud835\udc3asuperscriptvol\u25c6\ud835\udc3asuperscriptvol\u25c6superscript\ud835\udc3a2\ud835\udf0bm\ud835\udc5d\ud835\udc67\ud835\udc642\ud835\udf0bsubscriptsuperscript\ud835\udc67fd\ud835\udc3a{\\rm vol}^{\\perp}(G)={\\rm vol}(G)={\\rm vol}^{\\lozenge}(G)+{\\rm vol}^{\\lozenge}%\n(G^{*})=2\\pi\\,\\mathrm{m}(p(z,w))=2\\pi\\,z^{\\rm fd}_{G}.roman_vol start_POSTSUPERSCRIPT \u27c2 end_POSTSUPERSCRIPT ( italic_G ) = roman_vol ( italic_G ) = roman_vol start_POSTSUPERSCRIPT \u25c6 end_POSTSUPERSCRIPT ( italic_G ) + roman_vol start_POSTSUPERSCRIPT \u25c6 end_POSTSUPERSCRIPT ( italic_G start_POSTSUPERSCRIPT \u2217 end_POSTSUPERSCRIPT ) = 2 italic_\u03c0 roman_m ( italic_p ( italic_z , italic_w ) ) = 2 italic_\u03c0 italic_z start_POSTSUPERSCRIPT roman_fd end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_G end_POSTSUBSCRIPT .\n\n\n\nThus, the lower bound in Conjecture\u00a01 holds with equality.",'
    sentences:
      - '        "Let F\ud835\udc39Fitalic_F denote a target model, which will now be trained on a modified dataset Dp\u2062o\u2062i\u2062s\u2062o\u2062n\u2062e\u2062d=D\u2217subscript\ud835\udc37\ud835\udc5d\ud835\udc5c\ud835\udc56\ud835\udc60\ud835\udc5c\ud835\udc5b\ud835\udc52\ud835\udc51superscript\ud835\udc37D_{poisoned}=D^{*}italic_D start_POSTSUBSCRIPT italic_p italic_o italic_i italic_s italic_o italic_n italic_e italic_d end_POSTSUBSCRIPT = italic_D start_POSTSUPERSCRIPT \u2217 end_POSTSUPERSCRIPT, where D\u2217superscript\ud835\udc37D^{*}italic_D start_POSTSUPERSCRIPT \u2217 end_POSTSUPERSCRIPT is a surreptitiously modified version of the clean training dataset D\ud835\udc37Ditalic_D. The aim of data poisoning F\ud835\udc39Fitalic_F is creating a poisoned model F\u2217superscript\ud835\udc39F^{*}italic_F start_POSTSUPERSCRIPT \u2217 end_POSTSUPERSCRIPT that makes incorrect predictions, often without an observable degradation in its overall accuracy. Data poisoning compromises the model integrity by introducing systematic biases that serve the attacker\u2019s objectives while evading detection during model training.",'
      - '        "Figure 2 illustrates a comparison between the observed low-medium resolution and the high-resolution spectral profiles of the oxygen A band, depicting observations of (telluric) molecular oxygen. The upper panel of Figure 2 shows low to medium resolution telluric oxygen features. These were obtained from the ESO Science Archive Facility using X-shooter[141] observations during February and March 2024 by the UVES team, as part of Program ID: 60.A-9022(c), OB ID:2024672, 2024624 and 2024822, at various resolutions with short exposures (12 seconds). The results indicate that higher resolution enables the observation of more detailed features within the molecular oxygen spectrum, revealing the signal more distinctly within each spectral line. The lower panel of Figure 2 shows performance tests for future HRS instrumentation by observing the Sun through the Earth\u2019s atmosphere. These profiles demonstrate the measurement outcomes obtained using two types of interferometers: Michelson-based and FPI-based. Firstly, the FTS from the National Solar Observatory at Kitt Peak [126] reported R=700,000 in the oxygen A-band. Secondly, the FIOS-demo[133] showcases spectral profiles based on a chained FPI array with a spectral resolution of R=250,000. This resolution can potentially increase up to R=350,000 with the addition of each array. The throughput of each additional unit, however, decreases by 10-15% [50]. One benefit of achieving this level of resolution is the increase in signal-to-noise ratio and the sampling frequency for each spectral line, which may reduce the required observing time, as predicted in [46, 93].",'
      - '        "At this point, we can reconcile what we observe with the evidence from the last paragraphs on TFP in Figure 5. We argue that a critical mass is needed in either case to record a significant impact of the exporting activity. At lower levels of exporting activity, the company starts to benefit from economies of scale but also needs to invest in productive capacity. To keep up with the technological frontier is costly, and it often requires an upgrade of obsolete tangible assets. We argue that the combined evidence of rising operational capacity (sales and costs) and investment in fixed assets explains why we observe a negative albeit small productivity loss in an intermediate range of export intensity. It is only when the company operates abroad at a larger scale that positive albeit small TFP gains come as a consequence of exporting. In this case, we argue, economies of scale become evident and the capital adjustment unveils its impact on firms\u2019 performance.",'
  - source_sentence: '        "To generate queer warmth phrases, we employed persona prompting to adapt our SAE warmth phrases (see Table\u00a04). Three distinct personas were designed and used as prompts to produce iterations of the 14 SAE warmth phrases. Each phrase was processed through all three persona prompts (see Table\u00a08), resulting in a total of 42 unique queer warmth phrases. The final set of phrases is presented below.",'
    sentences:
      - '    "title": "Always skip attention",'
      - '        "To generate queer warmth phrases, we employed persona prompting to adapt our SAE warmth phrases (see Table\u00a04). Three distinct personas were designed and used as prompts to produce iterations of the 14 SAE warmth phrases. Each phrase was processed through all three persona prompts (see Table\u00a08), resulting in a total of 42 unique queer warmth phrases. The final set of phrases is presented below.",'
      - '        "Assuming an adequately sized Bloom filter, the proportion of false positives is small, ensuring that XAcomsuperscriptsubscript\ud835\udc4b\ud835\udc34comX_{A}^{\\text{com}}italic_X start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT start_POSTSUPERSCRIPT com end_POSTSUPERSCRIPT and XBcomsuperscriptsubscript\ud835\udc4b\ud835\udc35comX_{B}^{\\text{com}}italic_X start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT com end_POSTSUPERSCRIPT are highly similar. This minimizes the occurrence of similar but non-identical buckets, thereby mitigating the redundancy issue inherent in bucketing. Furthermore, the use of bucketing not only detects false positives but also ensures convergence, addressing the limitation of Bloom filters alone. This combined approach is analogous to the RSync protocol, where Bloom filters act as the weak checksum and bucketing serves as the strong checksum.",'
pipeline_tag: sentence-similarity
library_name: sentence-transformers

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

This is a sentence-transformers model finetuned from 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
Maximum Sequence Length: 256 tokens
Output Dimensionality: 384 dimensions
Similarity Function: Cosine Similarity

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': 256, '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:

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("Stergios-Konstantinidis/MNLP_M2_document_encoder")
# Run inference
sentences = [
    '        "To generate queer warmth phrases, we employed persona prompting to adapt our SAE warmth phrases (see Table\\u00a04). Three distinct personas were designed and used as prompts to produce iterations of the 14 SAE warmth phrases. Each phrase was processed through all three persona prompts (see Table\\u00a08), resulting in a total of 42 unique queer warmth phrases. The final set of phrases is presented below.",',
    '        "To generate queer warmth phrases, we employed persona prompting to adapt our SAE warmth phrases (see Table\\u00a04). Three distinct personas were designed and used as prompts to produce iterations of the 14 SAE warmth phrases. Each phrase was processed through all three persona prompts (see Table\\u00a08), resulting in a total of 42 unique queer warmth phrases. The final set of phrases is presented below.",',
    '    "title": "Always skip attention",',
]
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]

Training Details

Training Dataset

Unnamed Dataset

Size: 21,000 training samples
Columns: sentence_0, sentence_1, and label
Approximate statistics based on the first 1000 samples:
sentence_0 sentence_1 label
type string string int
details
min: 3 tokens
mean: 173.22 tokens
max: 256 tokens

min: 3 tokens
mean: 170.67 tokens
max: 256 tokens

0: ~66.60%
1: ~33.40%

	sentence_0	sentence_1	label
type	string	string	int
details	min: 3 tokens mean: 173.22 tokens max: 256 tokens	min: 3 tokens mean: 170.67 tokens max: 256 tokens	0: ~66.60% 1: ~33.40%

Samples:

sentence_0	sentence_1	label
"the user may robustify the design by selecting a suitable A^^\ud835\udc34\widehat{A}over^ start_ARG italic_A end_ARG. Only the choice of A^^\ud835\udc34\widehat{A}over^ start_ARG italic_A end_ARG has an impact at an algorithmic level and, normally, A^^\ud835\udc34\widehat{A}over^ start_ARG italic_A end_ARG is tuned to a set A\ud835\udc34Aitalic_A that, in the user\u2019s mind, captures, and suitably describes, possible adversarial actions. Still, we remark that our results hold true for any choice of A^^\ud835\udc34\widehat{A}over^ start_ARG italic_A end_ARG and A\ud835\udc34Aitalic_A (with A^\u2286A^\ud835\udc34\ud835\udc34\widehat{A}\subseteq Aover^ start_ARG italic_A end_ARG \u2286 italic_A), so accommodating situations in which, e.g., the user envisages adversarial actions of a certain type and, yet, he is willing to theoretically test the robustness of the design against actions of higher magnitude. One simple example of this situation occurs when the design is done...	"the user may robustify the design by selecting a suitable A^^\ud835\udc34\widehat{A}over^ start_ARG italic_A end_ARG. Only the choice of A^^\ud835\udc34\widehat{A}over^ start_ARG italic_A end_ARG has an impact at an algorithmic level and, normally, A^^\ud835\udc34\widehat{A}over^ start_ARG italic_A end_ARG is tuned to a set A\ud835\udc34Aitalic_A that, in the user\u2019s mind, captures, and suitably describes, possible adversarial actions. Still, we remark that our results hold true for any choice of A^^\ud835\udc34\widehat{A}over^ start_ARG italic_A end_ARG and A\ud835\udc34Aitalic_A (with A^\u2286A^\ud835\udc34\ud835\udc34\widehat{A}\subseteq Aover^ start_ARG italic_A end_ARG \u2286 italic_A), so accommodating situations in which, e.g., the user envisages adversarial actions of a certain type and, yet, he is willing to theoretically test the robustness of the design against actions of higher magnitude. One simple example of this situation occurs when the design is done...	`1`
"Aha Moment of R1-Reward. Through our task design and reward function formulation, the R1-Reward model effectively learns the reward modeling task structure during the SFT phase. Following reinforcement learning, it reduces the length of reasoning to enhance efficiency. Visual examples of the model\u2019s output appear in Figures\u00a03 and\u00a06. The model autonomously learns a process to assess response quality. It first defines the goal, analyzes the image, attempts to solve the problem, and provides an answer. Based on this, the model evaluates Response 1 and Response 2, compares the two outputs, and gives a final ranking. Simultaneously, the model demonstrates different reflection patterns. In Figure\u00a03, the model encounters an error in its calculation, but after rechecking the bar chart, it recognizes the mistake and recalculates to obtain the correct result. In Figure\u00a06, the model misunderstands the problem. However, after outputting \u201cWait, re-reading the ...	`"In an ideal case, the hole made after the punch doesn\u2019t move and keeps the size of the needle. Then the hole is filled with a subsequent paint layer, if it is not made in the top layer.",`	`0`
`"In our search for the optimal parameters, we evaluated all possible combinations presented in Section\u00a03.3. To do this, we aggregated the results for each specific parameter configuration and computed the mean metrics. This approach allowed us to isolate the effects of each parameter under evaluation.",`	`"We employ RWP to model the movement of humans within the indoor space and use the Matern hard-core process (MHCP) to model static obstacles, such as furniture or static humans, in the environment [15].",`	`0`

Loss: ContrastiveTensionLoss

Training Hyperparameters

Non-Default Hyperparameters

per_device_train_batch_size: 3
per_device_eval_batch_size: 3
num_train_epochs: 10
multi_dataset_batch_sampler: round_robin

All Hyperparameters

Click to expand

overwrite_output_dir: False
do_predict: False
eval_strategy: no
prediction_loss_only: True
per_device_train_batch_size: 3
per_device_eval_batch_size: 3
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: 10
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

Training Logs

Click to expand

Epoch	Step	Training Loss
0.0714	500	1.8871
0.1429	1000	1.7445
0.2143	1500	1.7138
0.2857	2000	1.699
0.3571	2500	1.6729
0.4286	3000	1.6864
0.5	3500	1.6718
0.5714	4000	1.6754
0.6429	4500	1.6747
0.7143	5000	1.6709
0.7857	5500	1.6797
0.8571	6000	1.6768
0.9286	6500	1.6694
1.0	7000	1.6754
1.0714	7500	1.6632
1.1429	8000	1.6643
1.2143	8500	1.6553
1.2857	9000	1.6626
1.3571	9500	1.6734
1.4286	10000	1.673
1.5	10500	1.6611
1.5714	11000	1.671
1.6429	11500	1.6762
1.7143	12000	1.6717
1.7857	12500	1.6599
1.8571	13000	1.681
1.9286	13500	1.6715
2.0	14000	1.6815
2.0714	14500	1.6304
2.1429	15000	1.6351
2.2143	15500	1.648
2.2857	16000	1.6538
2.3571	16500	1.6396
2.4286	17000	1.632
2.5	17500	1.6497
2.5714	18000	1.6526
2.6429	18500	1.6346
2.7143	19000	1.6548
2.7857	19500	1.6549
2.8571	20000	1.6438
2.9286	20500	1.6448
3.0	21000	1.6435
3.0714	21500	1.589
3.1429	22000	1.6075
3.2143	22500	1.6084
3.2857	23000	1.6061
3.3571	23500	1.6121
3.4286	24000	1.6168
3.5	24500	1.6022
3.5714	25000	1.6164
3.6429	25500	1.6132
3.7143	26000	1.6036
3.7857	26500	1.6077
3.8571	27000	1.6241
3.9286	27500	1.6224
4.0	28000	1.6023
4.0714	28500	1.5479
4.1429	29000	1.5569
4.2143	29500	1.5611
4.2857	30000	1.5413
4.3571	30500	1.5568
4.4286	31000	1.5458
4.5	31500	1.5405
4.5714	32000	1.5707
4.6429	32500	1.557
4.7143	33000	1.5561
4.7857	33500	1.5698
4.8571	34000	1.546
4.9286	34500	1.5589
5.0	35000	1.5692
5.0714	35500	1.5029
5.1429	36000	1.5087
5.2143	36500	1.4882
5.2857	37000	1.5116
5.3571	37500	1.5016
5.4286	38000	1.4988
5.5	38500	1.5065
5.5714	39000	1.5089
5.6429	39500	1.5104
5.7143	40000	1.4937
5.7857	40500	1.4974
5.8571	41000	1.5095
5.9286	41500	1.5064
6.0	42000	1.5119
6.0714	42500	1.4572
6.1429	43000	1.4732
6.2143	43500	1.4534
6.2857	44000	1.4598
6.3571	44500	1.4626
6.4286	45000	1.4486
6.5	45500	1.4677
6.5714	46000	1.4705
6.6429	46500	1.4757
6.7143	47000	1.4724
6.7857	47500	1.4744
6.8571	48000	1.4571
6.9286	48500	1.4571
7.0	49000	1.4549
7.0714	49500	1.4198
7.1429	50000	1.4328
7.2143	50500	1.4322
7.2857	51000	1.4191
7.3571	51500	1.4355
7.4286	52000	1.4409
7.5	52500	1.4366
7.5714	53000	1.4378
7.6429	53500	1.4229
7.7143	54000	1.4386
7.7857	54500	1.453
7.8571	55000	1.419
7.9286	55500	1.4215
8.0	56000	1.4248
8.0714	56500	1.4184
8.1429	57000	1.4059
8.2143	57500	1.4011
8.2857	58000	1.3962
8.3571	58500	1.4134
8.4286	59000	1.4104
8.5	59500	1.3924
8.5714	60000	1.4062
8.6429	60500	1.4117
8.7143	61000	1.4192
8.7857	61500	1.402
8.8571	62000	1.3998
8.9286	62500	1.4087
9.0	63000	1.4203
9.0714	63500	1.389
9.1429	64000	1.4049
9.2143	64500	1.3897
9.2857	65000	1.3839
9.3571	65500	1.3712
9.4286	66000	1.3908
9.5	66500	1.3986
9.5714	67000	1.4014
9.6429	67500	1.3919
9.7143	68000	1.404
9.7857	68500	1.3921
9.8571	69000	1.3918
9.9286	69500	1.4046
10.0	70000	1.3923

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: 3.6.0
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",
}

ContrastiveTensionLoss

@inproceedings{carlsson2021semantic,
    title={Semantic Re-tuning with Contrastive Tension},
    author={Fredrik Carlsson and Amaru Cuba Gyllensten and Evangelia Gogoulou and Erik Ylip{"a}{"a} Hellqvist and Magnus Sahlgren},
    booktitle={International Conference on Learning Representations},
    year={2021},
    url={https://openreview.net/forum?id=Ov_sMNau-PF}
}