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b1300d41c97b7f25bd15ee13d378364c09bd1dea | subsection | 38 | 56 | Minimizing a Polynomial ALS | To illustrate the main idea we (partially) minimize
a non-minimal “almost” polynomial
ALS \mathcal {A} = (u,A,v) of dimension n=6
for p = -xy + (xy + z). Note that we do not need knowledge
of the left and right family at all.
Let\mathcal {A} = \left(
\begin{bmatrix}
1 & . & . & . & . & .
\end{bmatrix},
\begin{bmatrix}
... | {
"cite_spans": []
} | 1809.05425 | Free Fractions: An Invitation to (applied) Free Fields | [
"Konrad Schrempf"
] | [
"math.RA"
] | 2,018 | en | Mathematics | [
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8bd7a8b9353ece7a5aa96b77ea92bcdb81ab8083 | subsection | 39 | 56 | Minimizing a Polynomial ALS | \\
. & 1 & T \\
. & . & I_{k+1:m}
\end{bmatrix}
\begin{bmatrix}
A_{1:,1:} & A_{1:,k} & A_{1:,:m} \\
. & 1 & A_{k,:m} \\
. & . & A_{:m,:m}
\end{bmatrix}
\begin{bmatrix}
I_{1:k-1} & . & . \\
. & 1 & U \\
. & . & I_{k+1:m}
\end{bmatrix} \\
&=
\begin{bmatrix}
A_{1:,1:} & A_{1:,k} & A_{1:,k} U + A_{1:,:m} \\
. & 1 & U + A_{... | {
"cite_spans": []
} | 1809.05425 | Free Fractions: An Invitation to (applied) Free Fields | [
"Konrad Schrempf"
] | [
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fd54d9b67053099efc4792d98d85742c15eeedb7 | subsection | 40 | 56 | Minimizing a Polynomial ALS | The blocks T= [\alpha _{k+1}, \alpha _{k+2}, \ldots , \alpha _n] and
U = [\beta _{k+1}, \beta _{k+2}, \ldots , \beta _n] in the transformation (P,Q)
are of size 1 \times (n-k), thus we have
a linear system of equations (over \mathbb {K}) with
2(n-k) unknowns (for k>1) and (d+1)(n-k) + 1 equations:\begin{bmatrix}
\beta ... | {
"cite_spans": []
} | 1809.05425 | Free Fractions: An Invitation to (applied) Free Fields | [
"Konrad Schrempf"
] | [
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d959bcd215f6ca45c3396700234a405ae2377a97 | subsection | 41 | 56 | Minimizing a Polynomial ALS | \\
. & . & I_{n-k}
\end{bmatrix}
\right)such that column k in PAQ is [0,\ldots ,0,1,0,\ldots ,0]^{\!\top }.
A sufficient condition for (t P^{-1})_k=0 is the existence of
T,U \in \mathbb {K}^{(k-1) \times 1} such thatA_{1:,1:} U + A_{1:,k} + T = 0.For the illustration we refer to
.
If a left (respectively right) minimiz... | {
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af5e52117ec1c0fd3cac78bdf62e4923b01ecce0 | subsection | 42 | 56 | Minimizing a Polynomial ALS | If we subtract row 3 from row 2 and add column 2 to column 3,
we get the ALS\mathcal {A}^{\prime } = (u^{\prime },A^{\prime },v^{\prime }) = \left(
\begin{bmatrix}
1 & . & . & . & .
\end{bmatrix},
\begin{bmatrix}
1 & -x & -x-y & x+y & . \\
. & 1 & 0 & . & 0 \\
. & . & 1 & . & -z \\
. & . & . & 1 & -y \\
. & . & . & . &... | {
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... | |
3fcd4e1c29e6c6365c5ee784c6440bbbcf67bb40 | subsection | 43 | 56 | Minimizing a Polynomial ALS | For k \in \lbrace 2,3,\ldots , n \rbrace the equations
A_{1:,1:} U + A_{1:,k} + T = 0, see (REF ),
with respect to the block decomposition \mathcal {A}^{[\underline{k}]}
are called right minimization equations,
denoted by \mathcal {R}_k = \mathcal {R}_k(\mathcal {A}). | {
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890d25078a534094e20340b49616357b7050430f | subsection | 44 | 56 | Minimizing a Polynomial ALS | A solution by the column block pair (T,U) is denoted by
\mathcal {R}_k(T,U) = 0,
the corresponding transformation by \bigl (P(T), Q(U) \bigr ).Algorithm 4.12 (Minimizing a polynomial ALS
)Input: \mathcal {A} = (u,A,v) polynomial ALS
of dimension n \ge 2 (for some polynomial p).Output: \mathcal {A}^{\prime } = (,,) if p... | {
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227e866743b794b4d49b882a97616867fc22e07f | subsection | 45 | 56 | Minimizing a Polynomial ALS | Notice that, compared to
, the first row does not have to be treated separately
(using an extended ALS), because for \dim \mathcal {A} =2
\mathbb {K}-linear independence of the left family is
equivalent to \mathbb {K}-linear independence of the right
family. Hence the former is indirectly checked by the
latter in line ... | {
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-0.005879981908947229,
-0.006811010651290417,... | |
ef4128dcf511bacd592cabc03a91a328ab597055 | subsection | 46 | 56 | Pivot Block Refinement | To be able to minimize an ALS using linear techniques
only the pivot blocks have to be refined,
that is, none can be (admissibly) transformed such that it
splits in two (smaller) pivot blocks.
For an illustration we consider the ALS\mathcal {A} = \left(
\begin{bmatrix}
1 & . & . & .
\end{bmatrix},
\begin{bmatrix}
1 & -... | {
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} | 1809.05425 | Free Fractions: An Invitation to (applied) Free Fields | [
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] | [
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0.0... | |
3883c2fc023d11b7af5e01dabfeedc0da25402e3 | subsection | 47 | 56 | Pivot Block Refinement | First we need to ensure invertibility of P and Q by
the conditions0 &\ne \det (P) = \alpha _{2,2}\alpha _{3,3} - \alpha _{2,3}\alpha _{3,2}
\quad \text{and}\\
0 &\ne \det (Q) =
(\beta _{2,2}\beta _{3,3} - \beta _{2,3}\beta _{3,2})\beta _{4,4} \\
& \qquad \qquad \qquad \qquad + (\beta _{2,4}\beta _{3,2} - \beta _{2,2}\b... | {
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e385a82a9e9c9b2078c5481db45fbebbf8d9d443 | subsection | 48 | 56 | Pivot Block Refinement | \end{bmatrix}
\begin{bmatrix}
\beta _{2,2} & \beta _{2,3} & \beta _{2,4} \\
\beta _{3,2} & \beta _{3,3} & \beta _{3,4} \\
\beta _{4,2} & \beta _{4,3} & \beta _{4,4}
\end{bmatrix}
&= \begin{bmatrix}
* & * & * \\
0 & * & * \\
0 & * & *
\end{bmatrix}
\quad \text{for $x$, and}\\
\begin{bmatrix}
\alpha _{2,2} & \alpha _{2,3... | {
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a6894ef1ea890702c4a6874a2cd6544bec000513 | subsection | 49 | 56 | Pivot Block Refinement | \\
. & 1 & 0 & 0 \\
. & 0 & 0 & \frac{1}{3} \\
. & -2 & 1 & 0
\end{bmatrix}
\right)yielding the (refined) admissible linear systemP \mathcal {A} Q = \left(
\begin{bmatrix}
1 & . & . & .
\end{bmatrix},
\begin{bmatrix}
1 & -z & . & . \\
. & x & 1 & . \\
. & . & y & -1 \\
. & . & -1 & x
\end{bmatrix},
\begin{bmatrix}
. \\... | {
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31e26b92b32a77604da76482ae67233b39e3d48b | subsection | 50 | 56 | Pivot Block Refinement | This approach is also recommended for the factorization of
polynomials (to create upper right blocks of zeros). | {
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d72805ea3472df614b92dbb0caca1171e82b194b | subsection | 51 | 56 | Minimizing a Refined ALS | The core of the minimization is to establish the
equivalence of minimality and the non-existence of
solutions of certain linear systems of equations.
Firstly we need to formalize what we have already done,
namely to apply (left and right) minimization steps
(as “solutions” to linear systems of equations).
This is somew... | {
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0c7b04e4388dde6e0ec5d7dba1f5fbe4a5c005ef | subsection | 52 | 56 | Minimizing a Refined ALS | \\
. & . & . & . & 1
\end{bmatrix}
\right)such that P\mathcal {A}Q has the form (“*” denotes an arbitrary entry)\begin{bmatrix}
1 & -x & z & * & * \\
. & 1 & -y & * & * \\
. & . & 1 & 0 & 0 \\
. & . & . & y & -1 \\
. & . & . & -z & x
\end{bmatrix}
s =
\begin{bmatrix}
. \\ . \\ 0 \\ . \\ 1
\end{bmatrix}by solving the li... | {
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} | 1809.05425 | Free Fractions: An Invitation to (applied) Free Fields | [
"Konrad Schrempf"
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e69374712eb152d1b66d797f42549ab828aa03b6 | subsection | 53 | 56 | Minimizing a Refined ALS | \\
. & . & . & 1
\end{bmatrix}
\right)such that P^{\prime }\mathcal {A}^{\prime }Q^{\prime } has the form\begin{bmatrix}
1 & -x & * & * \\
. & 1 & 0 & 0 \\
. & . & y & -1 \\
. & . & -z & x
\end{bmatrix}
s^{\prime } =
\begin{bmatrix}
. \\ 0 \\ . \\ 1
\end{bmatrix}.Since such a transformation exists (\alpha _{2,4}=\beta ... | {
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94eec9998a6931f1562a5f8237da11d3de6dd9c9 | subsection | 54 | 56 | Minimizing a Refined ALS | \\ 1
\end{bmatrix}.Notice that the entries \beta _{1,j} in the first row of Q^{\prime \prime } have to be
zero for (P^{\prime \prime },Q^{\prime \prime }) to be admissible and the corresponding entries in the left hand
side of\begin{bmatrix}
1 & 0 & 0
\end{bmatrix}
= t^{\prime \prime }
\begin{bmatrix}
1 & 0 & 0 \\
. & ... | {
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75bd2697c1bc86ddb6e0a62768a4d9bbc83fc91b | subsection | 55 | 56 | Epilogue | Learning to compute with fractions at school takes some time
and needs “hard” work by hand. This will not be different
for free fractions (but in general much more laborious).
For those who want to experiment in computer algebra systems:
An experimental implementation in
should be available soon. | {
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863b8fc27da05a32a89bd57f47ba43d23efabd80 | abstract | 0 | 45 | Abstract | We present a new machine learning approach to estimate personalized treatment
effects in the classical potential outcomes framework with binary outcomes. To
overcome the problem that both treatment and control outcomes for the same unit
are required for supervised learning, we propose surrogate loss functions that
inco... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
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78d81ccf77f917a4e396896100bec7393cd3a132 | subsection | 1 | 45 | Introduction | Many data-driven decisions, such as whether to prescribe a particular pharmaceutical drug or whether to launch a particular marketing campaign, are problems of causal inference that require conditional difference estimation. Causal inference considers the effects of interventions, which is the basis for policy-making. ... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
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d3ab7c69d4f769fc72f5af30892029e0c8cb2d9f | subsection | 2 | 45 | Introduction | Model complexity depends on the choice of kernel and regularization parameters, not on splitting or pruning parameters.One work that seems similar to ours on the surface but is not, is that of Ratkovic and Tingley , who use support vector machines (SVM's) only to determine the largest balanced subset of data, by classi... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
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ffb5dffca7c7a93af47e4c3bd8266d67d3748a12 | subsection | 3 | 45 | Problem Setting | We work in a standard potential outcomes setting, with observational data. Each observation possesses covariates and is assigned to either treatment or control groups, and an outcome is observed for each individual.
The potential outcomes for observation i are denoted by Y_i^T or Y_i^C, where the superscript T denotes ... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
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0a120974e23401dd940885ec5757a80afcb42a3b | subsection | 4 | 45 | Problem Setting | This is a relevant loss when we aim to correctly assign treatment to individual members of a population: e.g., optimally assigning advertisements to individuals visiting a website, or optimally assigning a pharmaceutical drug to individuals who would benefit from it.The second loss function that we consider isl_{\theta... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
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78504264b14b0aa52fcaea1c84ba7777c0c47490 | subsection | 5 | 45 | A Surrogate Conditional-Difference Loss Function | The following theorem defines sufficient conditions under which a surrogate loss function is valid for l_1.Theorem 1
If a function l(.) satisfies l(z) \ge \mathbb {1}_{z \ge 0} + \mathbb {1}_{z \ge 1}, then we have&\mathbb {E}_{X \sim \mu _{X|T},Y^T \sim \mu _{Y^T|X},Y^C\sim \mu _{Y^C|X}}l_{1}(X,Y^T,Y^C, h)
\\ &\le \m... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
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] | 2,018 | en | Statistics | [
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e1b959315e91f6c8bc396e324163a973f21fc0d7 | subsection | 6 | 45 | A Surrogate Conditional-Difference Loss Function | That sum would lead to separate modeling for treatment and control groups.One corollary of the theorem is a remark on the importance of accurate density ratio estimation. The following corollary shows that in some cases, it is not crucial to obtain an accurate estimate of the density ratio.Corollary 1 If for all functi... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
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0a8441a495c7d1eb3ed27d5562ee30da572cb223 | subsection | 7 | 45 | A Surrogate Conditional-Difference Loss Function | \lfloor 1+z \rfloor _+, the hinge loss function. We will use this loss function to construct an SVM-based algorithm.
(1+z)^2, the squared loss function.
\frac{2\ln (1+e^z)}{\ln (1+e)}, a scaled logistic loss function.
e^z, the exponential loss, used by AdaBoost. | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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7bd5b3ee1c22c998d416333bcd763625c515bacb | subsection | 8 | 45 | Conditional Difference SVM | In this section, we use the regularized hinge loss to formulate a quadratic programming problem that is similar to classical SVM (except that it is for potential outcomes data where we have only “half" of the label for each observation).For this section, we assume that the ratio \mu _C(x_i)/\mu _T(x_i) is either known ... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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5b5b951dd2dc9e8b983119506573caae8b951a7f | subsection | 9 | 45 | Conditional Difference SVM | \\&\left.\frac{1}{n^C}\ \sum _{i \in C} \frac{
\left\lfloor 1+(w_0+ K(w,x_i) )y_i^C\right\rfloor _+}{\mu _{X|C}(x_i) / \mu _{X|T}(x_i)} \right)+\gamma K(w,w).This minimax problem can be reformulated as a constrained optimization problem as follows:Primal Problem:{\min _{w,w_0,z,r,\forall i \;s_i,\forall i \; r_i} z+ \g... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
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374b14f4a331d0e0679000eb817caa5520f37d33 | subsection | 10 | 45 | Conditional Difference SVM | Its computational scaling properties are essentially identical to standard SVM.Recovering the Intercept w_0After solving for \lambda and \eta , we are able to theoretically recover an expression for \phi (w) in the primal formulation that can be used to obtain values of K(w,x) for any given x.
To make prediction possib... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
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101026a91293b2f288b416cbea8d35dac04ac63b | subsection | 11 | 45 | Conditional Difference SVM | Using that y_i is binary:w_0=y_i^T-K(w,x_i^T).Similarly, for i\in C if \eta _i < \frac{\beta }{n^C(\mu _{X|C}(x_i)/\mu _{X|T}(x_i))}, we conclude that s_i=0, and if for the same i \in C, \eta _i>0, we have
w_0=-y_i^C-K(w,x_i^C).
Also, using optimization methods that use a primal dual approach, it is possible to obtain ... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
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66c7a2d183b4c9cdebb9a47df0e5d0d86cd60b2c | subsection | 12 | 45 | Generalization Bound | The bound in this section provides a theoretical foundation for minimizing the maximum of treatment and control empirical errors.Definition 1 Growth Function: Let \mathcal {F} be a function class (also known as hypothesis class). Given data points z_1, \ldots , z_m, we consider \mathcal {F}_{z_1,\ldots , z_m}=\lbrace f... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
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ce19189efa77c6f71768d2be45277750b8e1db0e | subsection | 13 | 45 | Generalization Bound | We define a new loss function l^M(.) = \frac{1}{M}l(.).R_T(h)=\mathbb {E}_{X \sim \mu _{X|T},Y^T \sim \mu _{Y^T|X}} l^{M}(-h(X)Y^T).The corresponding empirical estimator for the expectation above would be\hat{R}_T(h)=\frac{1}{n_T}\sum _{i \in T} l^{M}(-h(x_i)Y^T).For the control group, we haveR_C(h)=\mathbb {E}_{X \sim... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
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6bc7e902a5d2edc2214cb2b34a07a90c7f76ec5e | subsection | 14 | 45 | Experiments | We cannot observe both treatment and control outcomes for the same observation in real data (this is not standard supervised learning), so ground truth treatment effects must be obtained another way for the purpose of evaluation.
In these experiments, the goal is to test the most basic potential outcomes setting. We ra... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
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1c7ed0178f3d655d83255d35d2074ec9e12f4c09 | subsection | 15 | 45 | Experiments | These are followed by matching based methods, difference of two supervised learning methods, and causal random forests. The two numbers for the causal random forest methods are the \alpha and \lambda parameters in that algorithm. The mean and the standard deviation (in braces) are reported in the table. The superscript... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
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20a5c9891f0f55524221f24be06fceca5ada7d49 | subsection | 16 | 45 | Breaking the Cycle of Drugs and Crime | Next, we apply our method to data from a social program in the United States, known as Breaking the Cycle (BTC), which studies the effect of intervention on the reduction of crime and drug use. These data were chosen for their relevance to treatment programs for the current opioid epidemic in the U.S. As far as we know... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
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"cs.LG"
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183ffdfc12bfb780b9aea887f1b8fe267a07c5da | subsection | 17 | 45 | Breaking the Cycle of Drugs and Crime | The rule list is below.if (have_drivers_license) then (effective) (87else if (long_term_serious_depression) then (not_effective) (20else if (long_term_trouble_understanding) then (effective) (92else if (SSI_benefit) then (effective) (100else if (prob_getting_along_with_father) then (effective) (91else (not_effective) (... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
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37eb26516f9a6fb0c85fb63701c2f84f0820a279 | subsection | 18 | 45 | Breaking the Cycle of Drugs and Crime | ACM, New York,NY, USA (2017).\newblock ISBN 978-1-4503-4887-4.\newblock Available from: \url{http://doi.acm.org/10.1145/3097983.3098047},\href {http://dx.doi.org/10.1145/3097983.3098047}{\path{doi:10.1145/3097983.3098047}}.\bibitem{athey2016recursive}\textsc{Athey, S. and Imbens, G.}\newblock Recursive partitioning for... | {
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} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
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"cs.LG"
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bdc80c4899a040c3c98b00b35ff71261f86dccdd | subsection | 19 | 45 | Breaking the Cycle of Drugs and Crime | 169--207.Springer (2004).\bibitem{cortes2010learning}\textsc{Cortes, C., Mansour, Y., and Mohri, M.}\newblock Learning bounds for importance weighting.\newblock In \emph{Advances in neural information processing systems}, pp.442--450 (2010).\bibitem{dehejia2002propensity}\textsc{Dehejia, R.~H. and Wahba, S.}\newblock P... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
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8d0e3359f2e73c495a7c120601a831b169c76e05 | subsection | 20 | 45 | Breaking the Cycle of Drugs and Crime | (2004).\newblock \url{https://doi.org/10.3886/ICPSR03928.v1}.\bibitem{hido2011statistical}\textsc{Hido, S., Tsuboi, Y., Kashima, H., Sugiyama, M., and Kanamori, T.}\newblock Statistical outlier detection using direct density ratio estimation.\newblock \emph{Knowledge and information systems}, \textbf{26} (2011), 309.\b... | {
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} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
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] | 2,018 | en | Statistics | [
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ea14f277fbd8b6170619b607f51f8873762805bf | subsection | 21 | 45 | Breaking the Cycle of Drugs and Crime | Stat.}, \textbf{7} (2013), 443.\newblock Available from: \url{https://doi.org/10.1214/12-AOAS593}, \href{http://dx.doi.org/10.1214/12-AOAS593} {\path{doi:10.1214/12-AOAS593}}.\bibitem{imai2013experimental}\textsc{Imai, K., Tingley, D., and Yamamoto, T.}\newblock Experimental designs for identifying causal mechanisms.\n... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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c229b7fbcf7bc0edc40dbb255a339ce9483fd258 | subsection | 22 | 45 | Breaking the Cycle of Drugs and Crime | Series A (General)},(1984), 656.\bibitem{rosenbaum1983assessing}\textsc{Rosenbaum, P.~R. and Rubin, D.~B.}\newblock Assessing sensitivity to an unobserved binary covariate in anobservational study with binary outcome.\newblock \emph{Journal of the Royal Statistical Society. | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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5da26f110f45e979144eb2305da5d82b369355eb | subsection | 23 | 45 | Breaking the Cycle of Drugs and Crime | Series B(Methodological)}, (1983), 212.\bibitem{Tianyu2017}\textsc{Roy, S., Rudin, C., Volfovsky, A., and Wang, T.}\newblock Flame: A fast large-scale almost matching exactly approach to causalinference.\newblock \emph{arXiv}, (2017).\bibitem{rubin1974estimating}\textsc{Rubin, D.~B.}\newblock Estimating causal effect... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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6acdce26ad8d1c65ae26ddad0a680b621de8347b | subsection | 24 | 45 | Breaking the Cycle of Drugs and Crime | Obtaining lower bounds for $\mathbb{E}_{X \sim \mu_{X|T},Y^T \sim \mu_{Y^T|X}} l(-h(X)Y^T)$ and $\mathbb{E}_{X \sim \mu_{X|C}, Y^C \sim \mu_{Y^C|X}} \frac{ l(h(X)Y^C) }{\mu_{X|C}(X)/\mu_{X|T}(X)}$.}\begin{align*}&\mathbb{E}_{X \sim \mu_{X|T},Y^T \sim \mu_{Y^T|X}, Y^C \sim \mu_{Y^C|X}} l(-h(X)Y^T) \\&= \mathbb{E}_{... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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6f1846484ce3615323ce9ec55a4a849fc042f64d | subsection | 25 | 45 | Breaking the Cycle of Drugs and Crime | Finding a lower bound for $\max \left( \mathbb{E}_{X \sim \mu_{X|T},Y^T \sim \mu_{Y^T|X}} l(-h(X)Y^T), \right. \\ \left. \mathbb{E}_{X \sim \mu_{X|C}, Y^C \sim \mu_{Y^C|X}} \frac{ l(h(X)Y^C)}{\mu_C(X)/\mu_T(X)}\right) $}We use the property that $a \geq b$ and $c \geq d$ imply $\max(a,c) \geq \max(b,d)$. Taking the ma... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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f9a247e97b4f77c1bfdca07be7053c50db7742b2 | subsection | 26 | 45 | Breaking the Cycle of Drugs and Crime | \\& + \mathbb{E}_{X \sim \mu_{X|T},Y^T \sim \mu_{Y^T|X}, Y^C \sim \mu_{Y^C|X}:Y^T<Y^C} \mathbbm{1}_{h(X)\geq 0} \left.+ \mathbb{E}_{X \sim \mu_{X|T},Y^T \sim \mu_{Y^T|X}, Y^C \sim \mu_{Y^C|X}:Y^T=Y^C} l(h(X)Y^C) \ \right) \\& \geq \mathbb{E}_{X \sim \mu_{X|T},Y^T \sim \mu_{Y^T|X}, Y^C \sim \mu_{Y^C|X}:Y^T>Y^C} \math... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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... | |
1a026bc10d884bc9b50ad413a1ec1fa3a61434a6 | subsection | 27 | 45 | Breaking the Cycle of Drugs and Crime | Lower bounds for $\mathbb{E}_{X \sim \mu_{X|T},Y^T \sim \mu_{Y^T|X}, Y^C \sim \mu_{Y^C|X}:Y^T=Y^C} l(-h(X)Y^T)$ and $\mathbb{E}_{X \sim \mu_{X|T},Y^T \sim \mu_{Y^T|X}, Y^C \sim \mu_{Y^C|X}:Y^T=Y^C} l(h(X)Y^C)$.}Since\begin{equation*}l(-h(X)Y^T) \geq 0\end{equation*}because it is an upper bound for an indicator functio... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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0... | |
6e6016da7adac2cc56f132a4142be7e51bf2475c | subsection | 28 | 45 | Breaking the Cycle of Drugs and Crime | Lower Bound for the maximum between \\$\mathbb{E}_{X \sim \mu_{X|T},Y^T \sim \mu_{Y^T|X}, Y^C \sim \mu_{Y^C|X}:Y^T=Y^C} l(-h(X)Y^T) $ and \\$\mathbb{E}_{X \sim \mu_{X|T},Y^T \sim \mu_{Y^T|X}, Y^C \sim \mu_{Y^C|X}:Y^T=Y^C} l(h(X)Y^C) $ }By using the fact that if $a \geq b$ and $c \geq d$, then we have $\max(a,c) \geq \... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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f6570fb9d246de2d940fdb288c18564ae62a9b98 | subsection | 29 | 45 | Breaking the Cycle of Drugs and Crime | \right) \\& \geq \mathbb{P}_{X \sim \mu_{X|T},Y^T \sim \mu_{Y^T|X}, Y^C \sim \mu_{Y^C|X}} (Y^T=Y^C=-1, h(X) \geq 1) \\& + \mathbb{P}_{X \sim \mu_{X|T},Y^T \sim \mu_{Y^T|X}, Y^C \sim \mu_{Y^C|X}} (Y^T=Y^C=1, h(X)\leq -1) \\ &+\mathbb{P}_{X \sim \mu_{X|T},Y^T \sim \mu_{Y^T|X}, Y^C \sim \mu_{Y^C|X}} (Y^T=Y^C=1, h(X) \... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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62bb6e018e23532aa35760e535a6fdde4e12ba83 | subsection | 30 | 45 | Breaking the Cycle of Drugs and Crime | Lower Bound for $\max ( \mathbb{E}_{X \sim \mu_{X|T},Y^T \sim \mu_{Y^T|X}} l(-h(X)Y^T) , \\ \mathbb{E}_{X \sim \mu_{X|C}, Y^C \sim \mu_{Y^C|X}} \frac{ l(h(X)Y^C) }{\mu_C(X)/\mu_T(X)})$}Combining the result from the seocnd step and fifth step, we have\begin{align*}&\max ( \mathbb{E}_{X \sim \mu_{X|T},Y^T \sim \mu_{Y^T... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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110a0b54f13575b855c978412948a24a15425233 | subsection | 31 | 45 | Breaking the Cycle of Drugs and Crime | \left. \mathbb{E}_{X \sim \mu_{X|C}, Y^C \sim \mu_{Y^C|X}} \frac{l(h(X)Y^C)}{\mu_{X|C}(X)/\mu_{X|T}(X)}\right). $$\end{proof}Remark: The proof of Theorem $2$ is actually included where we stop just before the final inequality.\begin{figure*}[htbp]\centering\begin{subfigure}[t]{0.35\textwidth}\centering\includegraphics... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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95335f8ed63d9e8bac69dc9da220a0b115822a65 | subsection | 32 | 45 | Breaking the Cycle of Drugs and Crime | We let $L_h(x)$ denote $L(h(x),f(x))$ in the absence of ambiguity about the target function $f$.For any hypothesis $h \in \mathcal{F}$, we denote by $R(h)$ its loss and by $\hat{R}_w(h)$ its weighted empirical loss:\begin{align*}R(h) &= \mathbb{E}_{x \sim P} [L(h(x)), f(x)] \\\hat{R}_w(h) &= \frac{1}{m}\sum_{i=1}^m w(x... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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1230c18523039293565802a316188795cca65330 | subsection | 33 | 45 | Breaking the Cycle of Drugs and Crime | Then, for any $\delta>0$, with probability at least $1-\delta$, the following holds:\begin{equation}\label{Cortes}\forall h \in \mathcal{F}, R(h) \leq \hat{R}_w(h) + 2^{\frac54} \sqrt{d_2(P||Q)}\sqrt[3/8]{\frac{p\log \frac{2ne}{p} + \log \frac{4}{\delta}}{n}}.\end{equation}From Equation \ref{Cortes}, we can conclude th... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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0c3e27e08d1abaf7b8aba8575b3b2318f029199f | subsection | 34 | 45 | Breaking the Cycle of Drugs and Crime | \end{align*}\subsection{Additional Experimental Results}Due to the page limit constraint in the main paper, here are results on some additional data sets.\subsubsection{Spiral Dataset without noise:}The first data set that we present is the spiral data set as shown in Figure \ref{coolspiral} without any noise. The supp... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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88b8be841681b69b24cb7c5fa024fff8673c72ea | subsection | 35 | 45 | Breaking the Cycle of Drugs and Crime | As we can see, our method is the best method without using difference of two supervised classifiers.}\label{fig:spirallosstheta}\end{table}\subsubsection{A Dataset Where the Treatment Effect Changes a Few Times}We construct a $2$-dimensional data set as follows. The features are distributed uniformly between $0$ and $1... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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2c294ca0a11226f43623eccef7cb49ac879797ed | subsection | 36 | 45 | Breaking the Cycle of Drugs and Crime | Numerical results are in Table \ref{Advantage}.\begin{table}[ht]\centering\small\begin{tabular}{rll}\hline& 0.01 & 0.1 \\\hlinelinear causal SVM 1e-8 & 62.1(3.18) & 55.3(3.3) \\linear causal SVM 1e-6 & 62.1(3.18) & 55.3(3.3) \\linear causal SVM 1e-4 & 61.5(3.34) & 54.6(3.06) \\quadratic causal SVM 1e-8 & 61.2(4.54) &... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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... | |
14747fb4b1bea708fed21fd1b724abbaf24a38c5 | subsection | 37 | 45 | Breaking the Cycle of Drugs and Crime | If the feature has norm that is bigger than $3$, then $y_i^{T}$ takes value $1$ with probability $0.8$ ; Otherwise, $y_i^T$ takes value $-1$ with probability $0.2$. $y_i^C$ always take value $1$ with probability $0.2$. Each data point has a probability of $0.3$ of being assigned to the control group.Table \ref{ControlT... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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92406697fa02eb65835fbb3d12a2e0e8cb7e10c6 | subsection | 38 | 45 | Breaking the Cycle of Drugs and Crime | It is shown that our RBF based methods beat matching based method.} \label{ControlTable}\end{table}\subsubsection{A dataset with treatment effect that changes a few times in high dimensions}This is a data set which consists of $1000$ data points where each data point consists of $120$ features. $60$ of the features fol... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
"Siong Thye Goh",
"Cynthia Rudin"
] | [
"stat.ML",
"cs.LG"
] | 2,018 | en | Statistics | [
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fb7248f7b863a7f922f9006911a4b9220beb89c3 | subsection | 39 | 45 | Breaking the Cycle of Drugs and Crime | It is equally likely for a data to be assigned to a treatment group or a control group.\begin{table}[ht]\centering\small\begin{tabular}{rll}\hline& 0.01 & 0.1 \\\hlinelinear causal SVM 1e-8 & 69.52(2.34) & 62.98(2.26) \\linear causal SVM 1e-6 & 69.5(2.38) & 62.98(2.26) \\linear causal SVM 1e-4 & 69.48(2.2) & 62.9(2.... | {
"cite_spans": []
} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
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a585049d8c24e2acfa79e52ed7c31ddcefdcff31 | subsection | 40 | 45 | Breaking the Cycle of Drugs and Crime | We also perform similar experiment under different assignment mechanism settings for this data set.\noindent \textbf{Setting 1: Equally likely to be assigned to be treatment or control group.}\begin{table}[ht]\centering\small\begin{tabular}{rll}\hline& 0.01 & 0.1 \\\hlinelinear causal SVM 1e-8 & 39.64(1.03) & 34.29(1.... | {
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} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
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04207cf68dfcf9bfabbcbd0f685edd76e66ae99e | subsection | 41 | 45 | Breaking the Cycle of Drugs and Crime | Causal SVM with RBF kernels performs similarly.\noindent \textbf{Setting 2: The assignment mechanism is based on $\operatorname{Bernoulli} \left( 0.75\left(\frac{1-\exp(-x_c^2)}{1+\exp(-x_c^2)}\right)\right)$}In this setting, we let our assignment mechanism be depending on the reading of citric acid. | {
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} | 1803.03769 | A Minimax Surrogate Loss Approach to Conditional Difference Estimation | [
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ebf914911270887e1d2f427cacdc8d9cf44c8753 | subsection | 42 | 45 | Breaking the Cycle of Drugs and Crime | We let $x_c$ denotes the reading of the citric acid and we let the probability that one is being assigned to the treatment group follows $\operatorname{Bernoulli} \left( 0.75\left(\frac{1-\exp(-x_c^2)}{1+\exp(-x_c^2)}\right)\right)$.\begin{table}[]\centering\small\begin{tabular}{rll}\hline& 0.01 & 0.1 \\\hlinelinear c... | {
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2d61ff38359769e398abf534bf7f27c1e0f82781 | subsection | 43 | 45 | Breaking the Cycle of Drugs and Crime | We let $x_c$ denotes the reading of the citric acid and we let the probability that one is being assigned to the treatment group follows $\operatorname{Bernoulli} \left( 0.5\left(\frac{1-\exp(-x_c^2)}{1+\exp(-x_c^2)}\right)\right)$.\begin{table}[]\centering\small\begin{tabular}{rll}\hline& 0.01 & 0.1 \\\hlinelinear c... | {
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aa72247d383be605d81ebae2ced745f9e0de3d73 | subsection | 44 | 45 | Breaking the Cycle of Drugs and Crime | Our experiments tend to favor the more complex model classes, such as difference of 2-SVM, despite the fact that there is no real theoretical principle underlying the use of 2-SVM. There are some advantages to using a single model, beyond the tighter bound on the 0-1 loss, and generalization bounds, in particular, bet... | {
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5faf50074d20bbfa247e05ee1419d6ce4b3d7a9f | abstract | 0 | 38 | Abstract | We introduce the notion of identity coercions between non-indexed and indexed
variants of inductive datatypes, such as lists and vectors. An identity
coercion translates one type to another such that the coercion function
definitionally reduces to the identity function. This allows us to reuse vector
programs to derive... | {
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} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
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3eb22a3a34c472d9d90b0f003cbb4e6fdb5dd316 | subsection | 1 | 38 | Introduction | In dependently typed languages
(such as Agda , Coq ,
Idris , or Lean )
it is common to define traditional algebraic datatypes,
as well as more refined indexed variants
of algebraic datatypes, where the values of the indexed type are a
restriction of the values of the original algebraic type to particular indices.
An ex... | {
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39ad564c420c6ed39c6c4b19acef28d30639c363 | subsection | 2 | 38 | The Setting | In a Curry-style type theory with implicit
products (such as ICC ),
an untyped Church-encoded vector can be assigned
the vector type (Vec),
but also the list type (List).
This is possible because the types share the same class of untyped
terms, and because vectors are a subtype of lists in ICC
(\textrm {Vec} \,\,A \,\,... | {
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"raw": "Miquel, A.: The implicit calculus of constructions extending pure type systems with an intersection type binder and subtyping. In: Internationa... | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
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dcb856f34517a7ca3825b9fadadd3dd67155eafb | subsection | 3 | 38 | Contributions | In \iota \lambda P2 , Stump adds a
dependent intersection type
and a heterogeneous equality type
to a type-annotated Curry-style calculus
with implicit products, allowing inductive types (i.e. those
supporting an induction principle) to be derived,
but whose erased terms are untyped Church-encodings.
Working in Cedi... | {
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a482dd2607ca6a3ce0d2dfe482efb3a65be0bd98 | subsection | 4 | 38 | Background: Deriving Inductive Types | In this section we review how to derive inductive types in
Cedille , whose erased terms are untyped
Church-encodings.
An inductive datatype is defined as the dependent intersection
of 3 components:The Church-encoding of the datatype.
The unary parametricity theorem of the Church-encoding.
The reflection theorem of th... | {
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d231e701ed0d2870f9bd7e7f45f3cef20cb175b3 | subsection | 5 | 38 | Church-Encoding | The first component (VecC) is the Church-encoded vector type,
where the impredicatively quantified
X is a family of types indexed by the natural numbers.Convention 1 We include “C” in
the suffix of an identifier to indicate that it relates to
a Church-encoded datatype.VecC ◂ ★ ➔ Nat ➔ ★ = λ A : ★ . λ n : Nat .∀ X : Nat... | {
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} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
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4d63e39a99c47c85c9aaa7158c29ef543d4aee1e | subsection | 6 | 38 | Church-Encoding | List · A ➔ ListC · B= Λ A . Λ B . λ f . λ xs . Λ X . λ cN . λ cC .xs.1.1 · X cN (λ x . cC (f x)) .Even though we have defined mapCL in abstract elimination
style, it is not an identity coercion (we know nothing about
the input function f, which may change the elements of the
list). However, if we partially apply mapCL ... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
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ef7ccc0dcd6d50dc0b92dde2e0b2baee014714e6 | subsection | 7 | 38 | Constructors | Next, we define the Church-encoded vector constructors nilCV
and consCV.Convention 2 We suffix an identifier with “V”
to indicate that it relates to vectors.nilCV ◂ ∀ A : ★ . VecC · A zero =Λ A . Λ X . λ cN . λ cC . cN .consCV ◂ ∀ A : ★ . ∀ n : Nat . A ➔ VecC · A n ➔ VecC · A (suc n) =Λ A . Λ n . λ x . λ xs . Λ X . λ c... | {
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... | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
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"Aaron Stump"
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d514aa175490d6006ea01ee05314bcd15379bb1e | subsection | 8 | 38 | Constructors | VecP · A n xsC ➔VecP · A (suc n) (consCV · A -n x xsC) =Λ A . Λ n . Λ xsC . λ x . λ xsP .Λ X . Λ P . Λ cN . Λ cC . λ pN . λ pC .pC -n -(xsC · X cN cC) x (xsP · X · P -cN -cC pN pC) .Any additional arguments that would get in the way of the
parametricity witnesses erasing to their corresponding
Church-encodings appear a... | {
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... | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
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"Aaron Stump"
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a79e4622573969b10c4467db86a3e32398bd6644 | subsection | 9 | 38 | Constructors | A ➔ Vec · A n ➔ Vec · A (suc n) =Λ A . Λ n . λ x . λ xs . mkVec · A -(suc n)[ consCV · A -n x xs.1.1 , consPV · A -n -xs.1.1 x xs.1.2 ]-(consRV · A -n -x -xs.1.1 -xs.2) .Below, we verify that the inductive constructors erase to their untyped
Church-encoded equivalents:\vert \textrm {nilV} \vert is the Church-encoding o... | {
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} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
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482454bdb1b34b9a1239d91a6f7554092aa47d07 | subsection | 10 | 38 | Unary Parametricity Theorem | The second component (VecP) is the unary parametricity
predicate on Church-encoded vectors (VecC). It takes 4 types
of abstract arguments, described below, and can be understood as an
abstract version of an eliminator (i.e. an induction
principle in type theory):An abstract return type (X).
An abstract motive (P, an a... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
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] | [
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682113617e29f9976a65e8f7b70a7f9b91f595a2 | subsection | 11 | 38 | Reflection Theorem | The third (and final) component (VecR) is the reflection
theorem for Church-encoded vectors. It states that eliminating a
vector as a vector, and using its constructors
(nilCV and consCV) for the branches, results in the
vector being eliminated:VecR ◂ Π A : ★ . Π n : Nat . VecC · A n ➔ ★ =λ A : ★ . λ n : Nat . λ xsC : ... | {
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f8cb755bbe40f137218d7e672ab6bf71d808f27b | subsection | 12 | 38 | Inductive Type | Finally, we define the inductive type of vectors (Vec) as the
dependent intersection (using type former \iota ) of the
Church-encoded vector type
(VecC from Section REF ) and its
parametricity theorem
(VecP from Section REF ), which is again
intersected with the reflection theorem for
Church-encoded vectors (VecR from ... | {
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-0.047002553939819336,
0.042241256684064865,
0.002380648860707879,
0.07044278830289841,
0.019258838146924973,
-0.01029325369745493,
-0.010903676971793175,
0.014604364521801472,
-0.... | |
fbd07b4b784b6bf6aa5f9b6dd93a5f91e01d0bc4 | subsection | 13 | 38 | Constructor Helper Function | Below, we define a helper function to construct a vector from the
intersection of VecC and VecP, and the reflection
theorem (VecR) as an implicit argument (➾ is syntax
for non-dependent ∀).mkVec ◂ ∀ A : ★ . ∀ n : Nat .Π xs : (ι xsC : VecC · A n . VecP · A n xsC) .VecR · A n xs.1 ➾ Vec · A n =Λ A . Λ n . λ xs . Λ q . [ ... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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... | |
8dc858e5324663d3d8e36b71f0d6a51c9a0cc45f | subsection | 14 | 38 | Eliminator | The whole point of defining the inductive vector type (Vec),
as opposed to the Church-encoded vector type (VecC), is so
we can define its eliminator (i.e. its induction principle in type theory):elimVec ◂ ∀ A : ★ . ∀ n : Nat . Π xs : Vec · A n .∀ P : Π n : Nat . Vec · A n ➔ ★ .Π pN : P zero (nilV · A) .Π pC : ∀ n : Nat... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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b975baf0604793da6d543d0d3bf489738aae1256 | subsection | 15 | 38 | Reusing Vector Definitions | In this section we demonstrate reusing vector programs and proofs to
define list-versions of the programs and proofs. Through the use of
identity coercions, our program reuse does not introduce runtime overhead,
and our proof reuse does not introduce equational reasoning overhead. | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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0.... | |
493b73641cfab79cf8dc94eb5f69c97122692209 | subsection | 16 | 38 | Identity Coercion from Vec to List | We extend Barras and Bernardo's identity
coercion from Church-encoded vectors to lists (v2lC),
to an identity coercion between inductive versions
of the types (v2l), i.e. those supporting induction
principles.
Identity coercions for inductive types are defined
using the same 3 components as inductive constructors (a
C... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-540-78499-9_26",
"end": 210,
"openalex_id": "https://openalex.org/W1554068457",
"raw": "Barras, B., Bernardo, B.: The implicit calculus of constructions as a programming language with dependent types. Foundations of Software S... | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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-0.012677960097789764,... | |
99701d788a6034bf7afbc64d3bab706f41ccff3d | subsection | 17 | 38 | Parametricity Theorem | Second, we translate the vector parametricity theorem to the list
parametricity theorem, this time projecting out the vector
parametricity theorem (via xs.1.2). In addition to the abstract
arguments that v2lC receives from its codomain,
v2lP also receives an abstract motive (P) and
abstract parametricity theorem branch... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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7ec55c099b20c47ab6d5365f3da8a019dd45b902 | subsection | 18 | 38 | Identity Coercion | Finally, we put together our 3 components
(v2lC, v2lP, and v2lR) to translate
inductive vectors to inductive lists, using the
mkList helper constructor. This is analogous to defining the
vector constructors in terms of their 3 components and mkVec
in Section REF .v2l ◂ ∀ A : ★ . ∀ n : Nat . Vec · A n ➔ List · A= Λ A . ... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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... | |
21decf892d5b0a0b7eaef1a6711f23eefddd8116 | subsection | 19 | 38 | Identity Coercion from List to Vec | Barras and Bernardo's non-dependent identity coercion takes
Church-encoded vectors to lists, which we have extended
(in Section REF ) to take inductive vectors to lists.
Because we are using inductive types, we can now define the
dependent identity coercion from inductive lists to vectors
(l2v). Church-encoded types ca... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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... | |
8455c9379df6a806e1af479c83ddb842a134a2a0 | subsection | 20 | 38 | Program Reuse | We achieve program reuse by defining list append
(appendL) in terms of vector append (appendV), in the
standard way by applying appendV to the result of coercing
both arguments to vectors (using l2v), and coercing the
result of vector append to a list (using v2l).appendL ◂ ∀ A : ★ . List · A ➔ List · A ➔ List · A= Λ A ... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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... | |
bd2f070c8af17cf1c7dfeb24cb8bd1cac15b8ac0 | subsection | 21 | 38 | Program Reuse | Vec · A m ➔Vec · A (add n m)= Λ A . Λ n . λ xs . Λ m . λ ys .ρ (v2u · A -n xs).2 -ρ (v2u · A -m ys).2 -ρ (lengthDistAppend · A (v2u · A -n xs).1 (v2u · A -m ys).1) -(l2v · A (appendL · A (v2u · A -n xs).1 (v2u · A -m ys).1)) .The result of reusing appendL has the following type:Vec · A (length (appendL (v2u xs).1 (v2u ... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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-0.03693167120218277... | |
5cf4b5fbcd89dd9a95dbaaced30ec61380020092 | subsection | 22 | 38 | Program Reuse | The result of this distribution is the product of the
length of the nested list and n:Vec · A (mult (length xss) n)Note that the property lengthDistConcat relies on all nested
lists having the same length (n), hence it is defined for a
list of length-constrained lists. Yet, our type resulting from reusing
concatL appli... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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... | |
b618d598d734db3340f11f5d3dc264c32a4a186a | subsection | 23 | 38 | Proof Reuse | Proof reuse, proving that list append is associative
(appendAssocL) in terms of a proof that
vector append is associative (appendAssocV),
is even easier than program reuse. We derive
appendAssocL by applying
appendAssocV to the result of coercing each argument from a
list to a vector (using l2v). We do not need to coer... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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-0.018123505637049675... | |
51a51f7bb9378a39c2aff36f425a7f8e5738a4de | subsection | 24 | 38 | Proof Reuse | Once again, this is easier than program reuse, as we must only
coerce the arguments to lists (using v2l), but must not
coerce the result (using l2v) because it is already an
equality type:appendAssocV ◂ ∀ A : ★ .∀ n : Nat . Π xs : Vec · A n .∀ m : Nat . Π ys : Vec · A m .∀ o : Nat . Π zs : Vec · A o .appendV (appendV x... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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0.0005111105274409056,
-0.00938307400792837... | |
e47923284ee5be548bba30f9b206d45291d358fe | subsection | 25 | 38 | Proof Reuse | This only requires applying our reused
proof of concatDistAppendL to the result of coercing
our input nested vector arguments to nested lists
(via v2l-v2l):concatDistAppendV ◂ ∀ A : ★ .∀ n1 : Nat . ∀ m1 : Nat . Π xss : Vec · (Vec · A n1) m1 .∀ n2 : Nat . ∀ m2 : Nat . Π yss : Vec · (Vec · A n2) m2 .appendV (concatV xss)... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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... | |
d866a04cfa35c3ad46f1b9277c9896a7d492bc18 | subsection | 26 | 38 | Reusing List Definitions | In this section we demonstrate reuse in the other direction
(compared to Section ), reusing list programs and proofs to
define vector-versions of the programs and proofs.
This direction of reuse takes more effort, because we may want to write
functions over vectors with index constraints in terms of functions
over list... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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... | |
9cb71a87e5d666a133e3381a6116d83ffc02c817 | subsection | 27 | 38 | Vectors as Length-Constrained Lists | The v2l function is “lossy” in the sense that the input
vector length does not appear in the list codomain.
In Section REF , we create a version of v2l
(named v2u) that “remembers” the index information, by
taking a vector to a list and a constraint on its length. Below, we
derive a new type (which will be the codomain... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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0.... | |
7d8cd22c654c31e5d84ae3b68f35fd160ff2ecf7 | subsection | 28 | 38 | Identity Coercion from Vec to VecL | Now we define v2u, taking a vector to a list and the constraint
that the length of the list is equal to the index of the vector
(by using VecL as the codomain of v2u). The function
v2u uses mkVecL to construct a VecL from a
vector by coercing to a list (via v2l), and proving the
constraint that v2l preserves the vector... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
-0.05784474313259125,
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... | |
929f49c239ddb23c125f3ad57aea9110bacd14fb | subsection | 29 | 38 | Reusing Nested List Definitions | In this section we demonstrate reuse for nested datatypes,
reusing programs and proofs over lists of lists
(List · (List · A)) to define programs and proofs
over vectors of vectors (Vec · (Vec · A n) m).
Like in Section , such reuse requires proving properties
about the lists to satisfy the vector length requirements o... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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0... | |
183d421f740ac1cd5070ab6b9854bcfcb231fba1 | subsection | 30 | 38 | List Map | To reuse a list of lists as a vector of vectors, we must be able to
coerce the inner lists in addition to the outer list. This can be
achieved by mapping v2l over the inner lists. However, to
ensure that reused definitions are identity coercions, it is crucial
that we define list map (mapL) in Barras and Bernardo's sty... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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0.01... | |
1251c92ceb0f7962446ec4560440955200ae7a98 | subsection | 31 | 38 | Nested Identity Coercions | In order to reuse a program over nested lists to derive a program
over nested vectors, we must coerce the nested vectors input of the
derived program to nested lists. Below, we define v2l-v2l
to perform such a coercion between nested datatypes.v2l-v2l ◂ ∀ A : ★ . ∀ n : Nat . ∀ m : Nat .Vec · (Vec · A n) m ➔ List · (Lis... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
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... | |
2c18947fff031f141464d0ee33b7d0e43a6d016b | subsection | 32 | 38 | Nested Identity Coercions | Thus, we also define the
nested mapping function v2u-v2l, which maps the outer vector
to a list, but remembers the inner list length constraints by mapping
the inner vectors to length-constrained lists (VecL):v2u-v2l ◂ ∀ A : ★ . ∀ n : Nat . ∀ m : Nat .Vec · (Vec · A n) m ➔ List · (VecL · A n)= Λ A . Λ n . Λ m . λ xss .... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
] | 2,018 | en | Computer Science | [
-0.03654937818646431,
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-0.0... | |
aa36f545e9b97c764f001f8da1ad9f54482a2ade | subsection | 33 | 38 | Identity Coercion from VecL to List | If we have a length-constrained list (VecL), we can retrieve
the inner list as the first projection of intersection:u2l ◂ ∀ A : ★ . ∀ n : Nat . VecL · A n ➔ List · A= Λ A . Λ n . λ xs . xs.1 .The nice thing about length-constrained lists is that they erase to
their list component, preventing the constraint from interfe... | {
"cite_spans": []
} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
] | [
"cs.PL"
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f1177b059ac9337f1392b704764e70ec0ec80d89 | subsection | 34 | 38 | Coercible in Haskell | Breitner et al. describe a GHC extension to Haskell (available
starting with GHC 7.8) for a type class Coercible a b, which
allows casting from a to b when such a cast is indeed
the identity function . The motivation is to support retyping of data
defined using Haskell's newtype statement, which is designed to
give pro... | {
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"raw": "Breitner, J., Eisenberg, R.A., Jones, S.P., Weirich, S.: Safe zero-cost coercions for Haskell. J. Funct. Program. 26, e15 (2016)",
"sourc... | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
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a9a5418dfb41d12abc07b97ff6379ff562f1bdc0 | subsection | 35 | 38 | Ornaments | Ornaments are used to define refined
version of types (e.g. Vec) from unrefined types
(e.g. List) by “ornamenting” the unrefined type with extra
index information. In contrast, our work establishes a relationship
between Vec and List after-the-fact, by defining
identity coercions in both directions for existing types.... | {
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"arxiv_id": "",
"doi": "",
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"source_ref_id": "a63c7ccf597cae8c3a0154fd7d05b0a5789c4b41",
"start": 0
},
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"a... | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
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c33429fdfcf958d521bbaf449b4a0a4a1eed7e73 | subsection | 36 | 38 | Type Theory in Color | Type Theory in Color (TTC)
generalizes the concept of erased arguments
of types to various colors, which may be erased optionally and
independently according to modalities in the type theory. In the vector
datatype declaration, the index data can be colored. If a vector is
passed to a function expecting a list (whose ... | {
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"doi": "10.1145/2500365.2500577",
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"raw": "Bernardy, J.P., Guilhem, M.: Type-theory in color. In: Proceedings of the 18th ACM SIGPLAN International Conference on Functional Programming. ... | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
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